# Quantum coherent microwave-optical transduction using high overtone bulk
acoustic resonances

Terence Blésin<EMAIL_ADDRESS>Institute of Physics, Swiss Federal
Institute of Technology Lausanne (EPFL), Lausanne, Switzerland  Hao Tian
OxideMEMS lab, Purdue University, West Lafayette, IN, USA  Sunil Bhave
OxideMEMS lab, Purdue University, West Lafayette, IN, USA  Tobias J.
Kippenberg<EMAIL_ADDRESS>Institute of Physics, Swiss Federal
Institute of Technology Lausanne (EPFL), Lausanne, Switzerland

###### Abstract

A device capable of converting single quanta of the microwave field to the
optical domain is an outstanding endeavour in the context of quantum
interconnects between distant superconducting qubits, but likewise can have
applications in other fields, such as radio astronomy or, in the classical
realm, microwave photonics. A variety of transduction approaches, based on
optomechanical or electro-optical interactions, have been proposed and
realized, yet the required vanishing added noises and an efficiency
approaching unity, have not yet been attained. Here we present a new
transduction scheme that could in theory satisfy the requirements for quantum
coherent bidirectional transduction. Our scheme relies on an intermediary
mechanical mode, a high overtone bulk acoustic resonance (HBAR), to coherently
couple microwave and optical photons through the piezoelectric and strain-
optical effects. Its efficiency results from ultra low loss and high
intracavity photon number sustaining integrated Si3N4 photonic circuits,
combined with the highly efficient microwave to mechanical transduction
offered by piezoelectrically coupled HBAR. We develop a quantum theory for
this multipartite system by first introducing a quantization method for the
piezoelectric interaction between the microwave mode and the mechanical mode
from first principles (which to our knowledge had not been presented in this
form), and link the latter to the conventional Butterworth-Van Dyke model. The
HBAR is subsequently coupled to a pair of hybridized optical modes from
coupled optical ring cavities via the strain-optical effect. We analyze the
conversion capabilities of the proposed device using signal flow graphs, and
demonstrate that quantum coherent transduction is possible, with realistic
experimental parameters at optical input laser powers of the order of
$\mathcal{O}(\text{mW})$. Combined with the high thermal conduction via the
device bulk, heating effects are mitigated, and the approach does not require
superconducting resonators that are susceptible to absorption of optical
photons.

##  I Introduction

Harnessing the effects of quantum mechanics is currently being investigating
with respect to producing technologies with higher performance than their
classical counter-parts in the domain of computing, sensing and communication,
and proves to be a formidable challenge in terms of the required engineering.
These improved performances of quantum technologies come at the expenses of
overcoming difficult obstacles [1]. Superconducting circuits are arguably one
of the most promising platforms for quantum computing owing the remarkably
high nonlinearity of Josephson junctions [2]. However, coupling between remote
qubits has become an important consideration in the development of
superconducting quantum computing on larger scales [3]. While superconducting
technologies enable consistently performing tasks in the quantum regime [4],
the GHz region of the electromagnetic spectrum they operate on is largely
populated by thermal excitations, requiring cryogenic temperatures to preserve
the quantum coherence of the logical units [5]. In contrast, optical signals
have a carrier frequency in the 100’s of THz region of the spectrum, with
negligible thermal noise even at room temperature, and quantum states of light
can be transported along optical fibers, with exceptionally low propagation
loss ($\sim$ 0.5 dB/km) that have enabled optical fibers to become the major
actor of high speed communication networks. One viable envisioned solution to
the scalability issue is to process information with superconducting circuits
at cryogenic temperature, and connect the different computing nodes with
optical photons via optical fibers at room temperature [3]. A quantum coherent
transducer, the type of device addressed in this article, would establish a
link between these two domains by coherently and bidirectionally converting
single photons at microwave and optical frequencies with high efficiency and
low added noise. In recent years, significant progress toward quantum coherent
microwave-to-optical transduction has been made, primarily using cavity
optomechanical, cavity electro-optical or piezooptomechanical conversion
schemes, i.e. processes relying on radiation pressure coupling [6], or the
electro-optical equivalent [7]. The concept of these approaches is illustrated
in the frequency domain on Fig. 1. However, achieving conversion efficiencies
approaching unity, which constitutes a stringent requirement to preserve
quantum correlations of the converted signals [8], remains an outstanding
engineering challenge, compounded by a variety of technical issues [9],
including e.g. impedance matching, optical fiber-chip coupling,
photorefractive effects, or even the vulnerability of superconductors to
optical photons [10]. While there are many ways to overcome these challenges,
it is nevertheless interesting to pursue the exploration of alternative
platforms for quantum coherent microwave-to-optical transduction, that
alleviate some of these shortcomings. In particular, it is opportune to
consider approaches based on wafer scale, high yield, ultra low loss
integrated photonic circuits, in particular based on Si3N4, which sustain
large intracavity photon numbers, and have shown quality factors $Q>30\times
10^{6}$ (i.e. propagation losses $\mathrm{<}$ 1dB/m) [11, 12].

The cavity optomechanical approach to microwave-optical conversion consists in
parametrically coupling a mechanical mode to both microwave and optical
cavities. Notable implementations include a Si3N4 membrane covered with Nb in
a Fabry-Pérot optical cavity [10, 13, 14] and Si photonic crystal zipper
cavities with Al [15]. This approach, although resorting to a low frequency
mechanical mode that reduces the conversion bandwidth to
$\mathcal{O}(\text{kHz})$, holds the record of measured efficiency of 40$\%$
for only 38 added noise photons [13].

For the cavity electro-optic approach, the microwave and optical fields are
directly coupled via Pockels’effect, and the resulting Hamiltonian is
equivalent to the optomechanical one [7]. Efficient conversion requires both
electric and optical fields to overlap in a material with broken
centrosymmetry, resulting in a three wave mixing process. Coupled racetracks
resonators made from LiNbO3 fabricated on SiO2 [16, 17] and sapphire [18, 19],
electrooptic polymer grown on Si waveguides [20], whispering gallery modes of
bulk LiNbO3 crystals [21], AlN single [22] and coupled rings optical cavities
[23], as well as Ti doped LiNbO3 phase modulators [24] all falls into this
category. These recent realizations have all relied on the use of coplanar
waveguides or lumped elements LC resonators [25] to significantly increase the
vacuum electric field strength, compared to bulk approaches [26, 7]. Yet,
despite impressive advances, the approach is compounded by photorefractive
effects limiting the available optical power and resonance frequency stability
[27].

The third approach also makes use of an intermediary excitation, similarly to
the first one, but the microwave is subsequently coupled to the mechanical
oscillator using a piezoelectric material. The auxiliary mechanical mode is
thus resonant with the microwave signal, which presents the advantage of
exhibiting larger conversion bandwidths, on the order of
$\mathcal{O}(\text{MHz})$. Moreover, piezoelectric materials efficiently
interconvert energy between electrical and mechanical forms, which earn them
an ubiquitous place in MEMS and wireless communication technologies [28]. To
complete the conversion process, the strain field of the mechanical mode
couples phonons to the optical mode. This method has been implemented with
suspended AlN beams patterned as optomechanical crystals [29], GaAs photonic
crystal nanobeams [30, 31], Si photonic crystal excited by AlN [32], LiNbO3
photonic crystal [33], LiNbO3 thin film acoustic resonator [34], optical and
acoustic whispering gallery modes of a LiNbO3 sphere [35], AlN microdisk [36].
Impedance matching between the piezoelectrically excited acoustic excitations
and mechanical modes hindering the conversion process, total conversion
efficiencies up to $\mathcal{O}(10^{-5})$ have been achieved [21] so far. The
transduction scheme presented in this paper falls in this last category. It
offers several unique advantages, including high intracavity photon number
handling capability, high conversion efficiency at realistic optical pump
powers, the use of bulk acoustic resonances rather than flexural modes of thin
film nanomechanical or micromechanical resonators, and being inherently
compatible with wafer scale manufacturing, such as the damascene process used
to fabricate ultra low loss Si3N4 photonic circuits [11]. Importantly, our AlN
actuator directly generates acoustic waves inside the HBAR cavity embedding
the optical Si3N4 waveguide. This configuration thus enables to circumvent the
problem of acoustic impedance matching that is severely limiting the SAW
excited optomechanical crystals [29, 31]. Combined with the release process to
step up the confinement of the HBAR mode in AlN, a material with high
piezoelectric coefficients, HBAR phonons efficiently fulfil their role as
intermediary between microwave and optical photons.

Our manuscript is organized as follows. We first describe the general idea of
the proposed device in section II. We then quantize the piezoelectric
interaction in section III in order to derive the correct coupling and decay
rates that describe the conversion of the itinerant microwave photon to a
phonon capable of interacting with the optical field, which to our knowledge
has not been carried out in this form before. We furthermore explain how to
experimentally measure these properties through microwave reflections. The
basic theory of radiation pressure effects in the proposed device is exposed
in section IV, together with simulations results justifying the expectation
regarding the conversion efficiency exposed in section VI. The formula for the
conversion efficiency of the proposed scheme is derived in section V, deriving
the Langevin equations for the modes of the transducer from the total
Hamiltonian, which deviate from prior treatments [17]. We introduce the use of
signal flow graphs to interpret the dependence of the conversion efficiency on
the system parameters, such as the cooperativities, in terms of feedback loops
between the different modes. Finally, we give in section VI an estimation of
performance of the proposed scheme for realistic values of the physical
parameters, obtained through simulations and confirmed by preliminary
measurements, corresponding to devices currently under fabrication.

##  II Description of the device

Figure 1:  Microwave-optical conversion schemes. (a) : (1) Schematic of the
schemes relying on the parametric coupling of both the optical cavity and MW
cavity on a mechanical mode. (2) Schematic of the schemes relying on
Pockels’effect to establish a direct link between MW and optics. (3) Schematic
of the schemes relying on piezoelectricity to couple the MW to a mechanical
mode acting on an optical cavity. This approach is the one described in this
paper. Contrarily to the first approach, it exhibits a high conversion
bandwidth as the mechanical frequency is matched to the electrical one. (b) :
Artistic rendering of the proposed device. The electrodes (in yellow and
orange) guide microwave photons to the piezoelectric material (in dark green)
to be converted into HBAR phonons. The silicon substrate (in light green) is
removed below the silicon oxide (in pink) HBAR cavity to reduce the mechanical
mode volume. This increases the interaction with optical photons in the
microring cavities (in blue). The inset in the top right of the figure
indicates that a HBAR phonon annihilates with an optical photon of the
symmetric supermode of the coupled rings to create an optical photon of the
asymmetric supermode. Conversely, an asymmetric supermode photon can be
annihilated to create a phonon.

This section presents the new type of devices discussed in this paper. The
transducer proposed in this paper consists of a pair of coupled optical ring
Si3N4 cavities in a SiO2 cladding, that serves as resonator for a high
overtone acoustic resonance (HBAR) addressed by a stack of electrodes and
piezoelectric material, for example AlN. The aforementioned device,
illustrated in Fig. 1, is the result of recent efforts on the combination of
AlN technology with silicon nitride integrated photonics that have led to the
successful demonstration of on-chip control over soliton microcombs [37], and
the realization of an integrated CMOS-compatible optical isolator [38].

The evanescent coupling between cavity modes of identical silicon nitride
micro-rings leads to the formation of effective supermodes, with a splitting
in the frequency domain that can be controlled with the device geometry. This
optical modes splitting is chosen to be equal to the frequency of the
microwave signal to convert. For transduction of microwave single photons, the
symmetric supermode, which has a lower frequency than the asymmetric one, is
pumped by an external laser and serves to convert photons between the
piezoelectric and the unpopulated asymmetric supermode. By using rings with a
small radius (circa 20 microns), the free spectral range of the optical cavity
approaches 1THz, and thus guarantees that the device can operate in a small
region of the optical spectrum containing only a pair of optical supermodes
[39, 40]. Furthermore, silicon nitride waveguides with very high intrinsic
quality factors ($Q>20\times 10^{6}$) have already been demonstrated [11].
This constitutes an important advantage as a slow decay rate of the optical
photons in the cavity lowers the input power threshold for optimal conversion.

The interfacing of the microwave signal to the optics is achieved by a
piezoelectric actuator (e.g. aluminium nitride) connected to the electrical
circuit, whose photons should be upconverted to the optical domain. The
voltage on the electrodes generates strains in the bottom layers, which leads
to acoustic waves generation for correct mechanical impedance matching. Here
we propose to etch part of the silicon layer beneath the silicon oxide
cladding to control the size of the mechanical cavity, enabling to control the
mechanical mass and frequency of the high overtone bulk acoustic wave. This is
a major difference with the modulators used in our previous work [38] as it
gives a boost of almost two orders of magnitude in optomechanical coupling,
and is expected to improve the quality factor of the HBAR resonator at low
temperatures [38, 37, 41, 42]. An important difference of the proposed device
with previous converters is that there is no microwave resonator : it is the
mechanical HBAR resonator itself that is overcoupled to the microwave signal
and gives it a resonant enhancement.

##  III Quantum mechanical description of the piezoelectric interaction

Figure 2:  Electromechanical conversion in the released HBAR device. (a) :
Cross-section of the actual device showing the stress pattern of the HBAR mode
excited by the electrodes at $3.285$GHz. The inset shows the optical mode
(norm of the electric field) in the optical ring waveguide. The white arrows
show the direction of the electric field, confirming that this mode is the
fundamental of the TE family. The small radius of the ring cavity ($22.5\
\mu$m) induces a shift of the mode from the waveguide center because of the
strong bending potential. (b) : Admittance curve of a released HBAR resonator
obtained by finite element simulations. Each pair of resonance and
antiresonance in the admittance $Y[\omega]$ corresponds to a mechanical
resonance, around which the BVD model can be applied to model the interaction
between the electric field and the mechanical motion. This curve shows two
strong HBAR resonances around $2.6$ GHz and $3.3$ GHz. The exact frequency of
the resonances and their optomechanical coupling rate depend on the position
of the waveguide inside the cladding. (c) : Signal flow chart to describe
microwave reflection measurements. It also justifies that near resonance we
can consider the travelling input microwave photon to be directly coupled to
the mechanics. (d) : Circuit diagram of the BVD model. It is used to model the
behaviour of piezoelectric materials around a mechanical resonance.

This section describes how a mechanical mode in the bulk beneath the actuator
exchanges energy with the electromagnetic field at microwave frequencies. In
contrast to the description of piezoelectric materials usually given in terms
of equivalent circuits [43, 44], we attempt to give an explanation in terms of
scattering amplitudes by starting from the microscopic relations in the
material and relying on energy conservation arguments. The advantage of this
procedure is that it directly provides a link to quantum Langevin equations
for solving the dynamics of the piezoelectric in the quantum regime, which has
not been found in the existing literature to the best of the authors knowledge
[43, 28]. We then discuss how the parameters appearing in this description can
be related to the Butterworth-Van Dyke model (BVD), commonly used in the MEMS
community [28] to characterize piezoelectric devices, and attempt to identify
these parameters in microwave reflection measurements.

###  III.1 Equations of motion

The aim of this subsection is to derive a coupling rate between microwave
photons and phonons in the HBAR cavity formed by the stack of electrodes,
piezoelectric, waveguide and cladding. For this purpose, we start from the
microscopic relations in piezoelectric materials [28]. Here we choose to use
stress-charge formulation, which states that in these materials a strain will
create free charges, and conversely an applied electric field will cause
additional stress.

The stress $T_{ij}$ in the piezoelectric material contains two contributions,
one from Hooke’s law, via the stiffness coefficients $c_{ijkl}$, and one from
the stress induced by an external electric field, via the piezoelectric stress
coefficients $e_{kij}$ [28]

$T_{ij}=c_{ijkl}S_{kl}-e_{kij}E_{k}$ (1)

Conversely, the electric displacement field in the piezoelectric material
contains an additional contribution which originates from strain induced free
charges

$D_{i}=\epsilon_{ik}E_{k}+e_{ikl}S_{kl}$ (2)

Further study of the piezoelectric interaction requires the Hamiltonian
density, which is derived from the Lagrangian $\mathcal{L}$ of the
corresponding fields following the procedure recalled in appendix A.

The electromagnetic energy density in the field generated by the electrodes,
including the piezoelectric, is given by

$\mathcal{H}_{\text{MW}}=\frac{1}{2}\left(\mathbf{E}\cdot\mathbf{D}\right)=\frac{1}{2}\left(\epsilon_{ij}E_{i}E_{j}+e_{ikl}E_{i}S_{kl}\right)$
(3)

We can clearly identify an additional contribution associated with the
coupling to the mechanical degree of freedom. The classical Hamiltonian is
obtained by integrating this energy density over the whole space. A new term,
only non-vanishing on the piezoelectric material, suggests the
electromechanical interaction term of the classical Hamiltonian to be

$H_{\text{int},\text{MW}}=\iiint_{\text{piezo}}\frac{1}{2}\left(e_{ikl}E_{i}S_{kl}\right)\
d\mathcal{V}$ (4)

Equivalently, we find that elastic waves in the released part of the device,
including the piezoelectric, contain an additional contribution to their
energy of

$H_{\text{int},m}=\iiint_{\text{piezo}}\frac{-1}{2}e_{kij}E_{k}S_{ij}\
d\mathcal{V}$ (5)

Having derived the interaction Hamiltonian from first principles, it is now
possible to quantize the Hamiltonian by first quantizing the fields
themselves. The mechanical displacement operator is defined as [45]

$\hat{\mathbf{u}}(\mathbf{r},t)=\sqrt{\frac{\hbar}{2m_{\text{eff}}\omega_{m}}}\mathbf{Q}(\mathbf{r})\hat{b}(t)+h.c.$
(6)

where $\hat{b}(t)$ is the annihilation operator describing the temporal
evolution of the mechanical mode and will be discussed in the rest of the
paper, $m_{\text{eff}}$ is the effective mass of the mode, related to the
stiffness in the different materials encountered by the acoustic wave, and
$\omega_{m}$ the eigenfrequency the mechanical mode. The mode function
$\mathbf{Q}(\mathbf{r})$ is a solution of the spatial part of the acoustic
wave equation for the conditions given by the geometry and materials of the
device considered

$-\omega_{m}^{2}Q_{i}(\mathbf{r})=\frac{\partial}{\partial
x_{j}}\left[c_{ijkl}(\mathbf{r})Q_{k,l}(\mathbf{r})\right]$ (7)

using Einstein’s notation for conciseness. This mode function, corresponding
to an adimensional displacement function, is normalized such that

$\iiint_{\text{HBAR}}\rho(\mathbf{r})\mathbf{Q}^{*}(\mathbf{r})\cdot\mathbf{Q}(\mathbf{r})\
d\mathcal{V}=m_{\text{eff}}$ (8)

to remain consistent with the definition given for the displacement operator.
The prefactor in front of the spatial and temporal parts gives the
displacement fluctuations of the zero-point of motion
$u_{\text{ZPF}}=\sqrt{\frac{\hbar}{2m_{\text{eff}}\omega_{m}}}$. The strains
are then obtained from the displacement operator
$S_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i})$.

Similarly the quantized form of the electric field relevant in this setting of
piezoelectric interaction is [46]

$\hat{\mathbf{E}}(\mathbf{r},t)=-\sqrt{\frac{\hbar\omega_{\text{MW}}}{2\epsilon_{\text{eff}}}}\mathbf{F}(\mathbf{r})\hat{c}(t)+h.c.$
(9)

with an annihilation operator $\hat{c}(t)$ for the temporal evolution that
will be formally eliminated from the description in a later section, an
effective permeability seen by the mode $\epsilon_{\text{eff}}$ and a
particular mode function $\mathbf{F}(\mathbf{r})$, solution of the spatial
part of Helmholtz equation

$\nabla\times\nabla\times\mathbf{F}(\mathbf{r})=\frac{\epsilon(\mathbf{r})}{\epsilon_{0}}\frac{\omega_{\text{MW}}^{2}}{c^{2}}\mathbf{F}(\mathbf{r})$
(10)

normalized such that

$\iiint_{\text{whole
space}}\frac{\epsilon(\mathbf{r})}{\epsilon_{0}}\mathbf{F}^{*}(\mathbf{r})\cdot\mathbf{F}(\mathbf{r})\
d\mathcal{V}=\epsilon_{r,\text{eff}}$ (11)

to define the effective relative permittivity.

The quantum mechanical Hamiltonian for the piezoelectric interaction with the
microwave field is then obtained by inserting these quantized displacement (6)
and microwave (9) fields in equation (4)

$\hat{H}_{\text{int},\text{MW}}=\frac{-1}{4}\hbar\sqrt{\omega_{\text{MW}}\omega_{m}}\underbrace{\left(\iiint_{\text{piezo}}\sqrt{\frac{e_{ikl}^{2}}{c_{\text{eff}}^{E}\epsilon_{\text{eff}}^{T}}}F_{i}(\mathbf{r})\frac{1}{2}[Q_{k,l}(\mathbf{r})+Q_{l,k}(\mathbf{r})]\
d\mathcal{V}\right)}_{\sqrt{k_{\text{eff}}^{2}}}\left(\hat{b}+\hat{b}^{\dagger}\right)\left(\hat{c}+\hat{c}^{\dagger}\right)$
(12)

When defining an effective stiffness
$c_{\text{eff}}^{E}=m_{\text{eff}}\omega_{m}^{2}$ for the mechanical mode and
an effective permittivity for the corresponding microwave mode $\epsilon^{T}$
at constant strain, we notice the emergence of the _electromechanical coupling
factor_ as defined by the MEMS community [47]

$K^{2}=\frac{e^{2}}{c^{E}\epsilon^{T}}$ (13)

The integral over the piezoelectric volume of this quantity then gives the
effective electromechanical coupling factor $\sqrt{k_{\text{eff}}^{2}}$, a
quantity easily extracted from the BVD model as explained in the next section.

Taking into account the fact that an identical term arise from the increase in
acoustic wave energy, the piezoelectric interaction Hamiltonian finally takes
the form

$\hat{H}_{\text{int}}=-\hbar
g_{\text{EM}}\left(\hat{b}+\hat{b}^{\dagger}\right)\left(\hat{c}+\hat{c}^{\dagger}\right)$
(14)

with a piezoelectric electromechanical coupling rate defined as

$g_{\text{EM}}=\frac{1}{2}\sqrt{k_{\text{eff}}^{2}}\sqrt{\omega_{m}\omega_{\text{MW}}}$
(15)

Importantly, the value of this coupling rate is independent of the amplitude
of the strain field and electric field, being only determined by the material
properties and geometry of the device. We note that this coupling rate has the
same form as the one derived by E. Zeuthen et al., although the derivation is
based on a different model [43, 30].

###  III.2 Butterworth-Van Dyke model

For experimental characterization of piezoelectrics, it is common to resort to
an equivalent circuit, which was conceived by Butterworth and Van Dyke [28].
While later modifications were brought to better fit experimental data, the
essence of the model lies in the parallel coupling between a branch containing
the behaviour of the electrodes in absence of piezoelectric effect with a
branch representing the motion of the mechanical degree of freedom, up to some
voltage conversion factor. The static capacitance $C_{0}$ takes into account
the usual capacitive behaviour of the electrodes, from the energy stored via
the permittivity of the material without the piezoelectric effect. The
motional branch connected in parallel arises from the supplementary charges
created on the electrodes by the mechanical deformation via the piezoelectric
effect. The motional capacitance $C_{m}$ accounts for the potential mechanical
energy while the motional inductance $L_{m}$ accounts for the kinetic
mechanical energy. Finally the dissipation of acoustic waves is modelled by a
motional resistance $R_{m}$. This equivalent circuit is illustrated in Fig. 2.
Piezoelectric devices are usually characterized by looking at their
admittance, which is ideally represented by the BVD model admittance

$Y(\omega)=i\omega
C_{0}+\frac{1}{L_{m}}\frac{i\omega}{-\omega^{2}+i\omega\gamma_{0}+\omega_{m}^{2}}$
(16)

near a resonance. $\omega_{m}$ and $\gamma_{0}$ are respectively the resonant
frequency and linewidth of the electrically probed mechanical resonance. A few
parameters determining the behaviour of the device and quality of the
mechanical resonance can be extracted from the curve of the admittance as a
function of the frequency. Each mechanical mode will exhibit a peak at what is
called a series resonance frequency $\omega_{s}$ and a dip at what is called a
parallel resonance frequency $\omega_{p}$. Away from the resonance the device
behaves as a capacitor of value $C_{0}$. The bandwidth of the conversion
between electromagnetic energy and mechanical energy is described by the
effective electromechanical coupling factor

$k_{\text{eff}}^{2}=\frac{\omega_{p}^{2}-\omega_{s}^{2}}{\omega_{p}^{2}}$ (17)

which also fixes the value of the electromechanical coupling rate. This
coupling factor then indicates the scale of the motional capacitance compared
to the static effect

$C_{m}=k_{\text{eff}}^{2}C_{0}$ (18)

The last parameters $L_{m}$ and $R_{m}$ are then extracted from the frequency
$\omega_{m}$ and linewidth $\gamma_{0}$ of the mechanical resonance.
111Comparing the microwave reflection spectrum and the admittance shows that
the mechanical resonance is in the middle of the series and parallel
resonances $\omega_{m}=\frac{\omega_{s}+\omega_{p}}{2}$.

$\displaystyle L_{m}$ $\displaystyle=\frac{1}{\omega_{m}^{2}C_{m}}$ (19)
$\displaystyle R_{m}$ $\displaystyle=L_{m}\gamma_{0}$ (20)

Fig. 2 shows the admittance curve of the HBAR resonator considered here. This
curve was obtained by finite elements simulations. Each mechanical resonance
features a succession of resonance and antiresonance on the admittance. The
height of the resonance indicates the strength of the coupling to the
microwave mode. While the simulated curve exhibits several resonant features,
two particular frequencies, corresponding to the silicon oxide HBAR modes, are
standing out near $2.6$GHz and $3.2$GHz. The latter one is the one considered
for the numerical evaluations presented in the rest of the paper as its
electromechanical coupling factor takes a higher value.

###  III.3 Link to MW reflection measurements

This section attempts to reconcile the engineers picture with the physicists
picture by showing how reflection measurements performed in laboratory are
linked to the annihilation operators description, which holds a central place
in second quantization. The admittance curve fitted by the BVD model as
described in the previous section is generally obtained via microwave
reflection measurements $S_{11}[\omega]$. These measurements are also easily
described in terms of annihilation operators. A single microwave mode
interacting with a single mechanical mode via the piezoelectric interaction

$\hat{H}_{\text{piezo}}=-\hbar
g_{\text{EM}}\left(\hat{b}+\hat{b}^{\dagger}\right)\left(\hat{c}+\hat{c}^{\dagger}\right)$
(21)

are evolving according to the coupled modes equations

$\displaystyle\frac{d}{dt}\hat{b}=$
$\displaystyle-i\omega_{m}\hat{b}-\frac{\gamma_{0}}{2}\hat{b}$
$\displaystyle+ig_{\text{EM}}\left(\hat{c}+\hat{c}^{\dagger}\right)+\sqrt{\gamma_{0}}\hat{f}_{m}$
(22) $\displaystyle\frac{d}{dt}\hat{c}=$
$\displaystyle-i\omega_{\text{MW}}\hat{c}-\frac{\Gamma}{2}\hat{c}$
$\displaystyle+ig_{\text{EM}}\left(\hat{b}+\hat{b}^{\dagger}\right)+\sqrt{\Gamma_{0}}\hat{f}_{\text{MW}}+\sqrt{\Gamma_{\text{ex}}}\hat{c}_{\text{in}}$
(23)

Taking Laplace’s transform 222Defined here and in the rest of the article such
that $f(t)\rightarrow f[s]=\int_{0}^{\infty}e^{-st}f(t)dt$ and defining
susceptibilities for the microwave field 333 $s$ is the complex frequency from
Laplace transform. Its imaginary part is related to the frequency with the
physicist convention as $\text{Im}(s)=-\omega$. This convention is used in the
rest of the paper to go from Laplace transform to the frequency domain.

$\chi_{\text{MW}}[s]=\frac{1}{s+i\omega_{\text{MW}}+\frac{\Gamma}{2}}$ (24)

and for the strain field

$\chi_{m}[s]=\frac{1}{s+i\omega_{m}+\frac{\gamma_{0}}{2}}$ (25)

the microwave reflection coefficient $S_{11}[\omega]$ is readily obtained from
the signal flow graph on Fig. 2 using Mason’s gain formula [48]

$S_{11}[s]=\frac{\hat{c}_{\text{out}}}{\hat{c}_{\text{in}}}=-1+\Gamma_{\text{ex}}\frac{\chi_{\text{MW}}[s]}{1+g_{\text{EM}}^{2}\chi_{\text{MW}}[s]\chi_{m}[s]}$
(26)

Finally, we wish to justify the form of the equation for the mechanics that
will be used in the section on the conversion efficiency of the transducer,
where the mechanical mode is considered to be directly coupled to the input
microwave photon, without having to go through an intermediary microwave mode
with an equation of the type

$\frac{d}{dt}\hat{b}=-i\omega_{m}\hat{b}-\frac{\gamma_{m}}{2}\hat{b}+\sqrt{\gamma_{\text{piezo}}}\hat{f}_{\text{piezo}}+\sqrt{\gamma_{\text{ex}}}\hat{c}_{\text{in}}$
(27)

to which the optomechanical interaction will be added. The main point is that
the internal dynamics of the microwave mode can be hidden to the optics, which
will effectively see the mechanics as if it was directly pumped by the
incoming microwave photons. This is done using the microwave mode
susceptibility to express it as a function of the other terms in the coupled
modes of the piezoelectric

$\displaystyle\hat{c}[s]$
$\displaystyle=i\chi_{\text{MW}}[s]g_{\text{EM}}\hat{b}[s]$
$\displaystyle+\chi_{\text{MW}}[s]\sqrt{\Gamma_{0}}\hat{f}_{\text{MW}}[s]+\chi_{\text{MW}}[s]\sqrt{\Gamma_{\text{ex}}}\hat{c}_{\text{in}}[s]$
(28)

which fixes the coefficients to use in the desired equation from the original
coefficients of the separate mechanical and microwave systems:

$\displaystyle\begin{cases}\frac{\gamma_{m}}{2}=\frac{\gamma_{0}}{2}+\chi_{\text{MW}}[s]g_{\text{EM}}^{2}\\\
\sqrt{\gamma_{\text{noise}}}\hat{f}_{\text{piezo}}=\sqrt{\gamma_{0}}\hat{f}_{m}+ig_{\text{EM}}\chi_{\text{MW}}[s]\sqrt{\Gamma_{0}}\hat{f}_{\text{MW}}\\\
\sqrt{\gamma_{\text{ex}}}=ig_{\text{EM}}\chi_{\text{MW}}[s]\sqrt{\Gamma_{\text{ex}}}\end{cases}$
(29)

Since the electrodes are directly connected to the source and that no
inductive element is added in the circuit, we can assume that the coupling
rate from the source to the microwave mode is directly given by the charging
time of the static capacitance $C_{0}$ by the characteristic impedance
$Z_{0}=50\Omega$

$\Gamma_{\text{ex}}=\frac{1}{Z_{0}C_{0}}$ (30)

Similarly the intrinsic decay rate of the microwave mode is given by the
discharge time of the static capacitance in the parallel resistance of the
modified BVD circuit shown in Fig. 2

$\Gamma_{0}=\frac{1}{R_{0}C_{0}}$ (31)

Simulations and preliminary experiments show that $C_{0}\approx 200\text{fF}$
and $R_{0}\approx\mathcal{O}(10\text{k}\Omega)$. For these values the
microwave mode has a total linewidth
$\Gamma=\Gamma_{0}+\Gamma_{\text{ex}}\approx\mathcal{O}(15\text{GHz})$ and is
strongly overcoupled to the source, with
$\frac{\Gamma_{\text{ex}}}{\Gamma}>95\%$. Under these conditions, the
effective linewidth of the mechanical mode addressed by the microwave source
is

$\gamma_{m}\approx\gamma_{0}+4\frac{g_{\text{EM}}^{2}}{\Gamma}$ (32)

and the microwave photon to phonon coupling rate is

$|\gamma_{\text{ex}}|\approx\frac{4g_{\text{EM}}^{2}}{\Gamma}\frac{\Gamma_{\text{ex}}}{\Gamma}$
(33)

The large linewidth and strong overcoupling of the microwave mode explain why
mechanical modes of piezoelectric resonators can be directly probed with
microwave reflection measurements, and why no mode splitting is observed
despite having an electromechanical coupling rate greater than the mechanical
linewidth.

##  IV Optomechanical interaction

In this section we move to the optical side of the device, which is interfaced
to the microwave part via the mechanics. The optomechanical interaction
essentially expresses the fact that the optical energy can be transferred to a
mechanical degree of freedom, and vice-versa. For the proposed device,
composed of a silicon nitride waveguide in silicon oxide cladding, this
phenomenon finds its root in two effects: either the optical field can act on
the mechanical field via radiation pressure and the latter reacts via moving
boundaries, or the optical field acts on the mechanical field via
electrostriction and the latter reacts by photoelasticity.

###  IV.1 Photoelastic effect

Strains in the material are changing the potential seen by the electrons,
which change the permittivity of the medium. This directly translate to a
change in refractive index, and this effect is called photoelasticity.

Mathematically, photoelasticity is easily expressed as a change in the
impermeability $\eta=\frac{1}{\epsilon_{r}}$

$\Delta\eta_{ij}=\sum_{kl}p_{ijkl}S_{kl}$ (34)

The modulation of impermeability obviously corresponds to a modulation in
relative permittivity

$(\Delta\epsilon_{r})_{ij}=\sum_{kl}-\epsilon_{r,ik}\epsilon_{r,kl}(\Delta\eta)_{lj}$
(35)

A change in permittivity corresponds to a variation of the energy stored in
the electromagnetic field, and thus a change in the optical frequency given by
Bethe-Schwinger formula for optical cavities perturbation [49]

$\Delta\omega=\frac{-\omega_{0}}{2}\frac{\iiint\expectationvalue{\Delta\epsilon}{E}d\mathcal{V}}{\iiint\expectationvalue{\epsilon}{E}d\mathcal{V}}$
(36)

The optomechanical coupling rate for a single photon depends on the cavity
frequency shifts and the zero point fluctuations motion of the quantity
causing the modulation, here the displacement in the waveguide and cladding
$g_{0}=\frac{\partial\omega}{\partial S}u_{\text{ZPF}}$.444Here we used
$S_{\text{ZPF}}=u_{\text{ZPF}}$. The contribution to the optomechanical single
photon coupling strength from photoelasticity is thus

$g_{0,\text{OM,PE}}=-\frac{\omega_{0}}{2}\frac{\expectationvalue{\mathbf{E}{\frac{\partial\epsilon}{\partial
S}}{\mathbf{E}}}}{\int\mathbf{E}\cdot\mathbf{D}\ d\mathcal{V}}u_{\text{ZPF}}$
(37)

###  IV.2 Moving boundaries effect

The deformation of the waveguide changes the confinement of the optical mode
in the different media, which in return modifies its effective refractive
index. The associated change in frequency is again computed using Bethe-
Schwinger formula for cavity perturbation.

$g_{0,\text{OM,MB}}=-\frac{\omega_{0}}{2}\frac{\oint(\mathbf{S}_{\text{ZPF}}\cdot\bar{\mathbf{n}})(\Delta\epsilon\mathbf{E}_{||}^{2}-\Delta\epsilon^{-1}\mathbf{D}_{\perp}^{2})d\mathcal{S}}{\int\mathbf{E}\cdot\mathbf{D}d\mathcal{V}}$
(38)

Here $\bar{\mathbf{n}}$ denotes the normal vector of the interface between
Si3N4 and SiO2, $\Delta\epsilon=\epsilon_{Si_{3}N_{4}}-\epsilon_{SiO_{2}}$ is
the permittivity difference between supporting media,
$\Delta\epsilon^{-1}=\epsilon_{Si_{3}N_{4}}^{-1}-\epsilon_{SiO_{2}}^{-1}$ is
the impermeability difference between the supporting media, $\mathbf{E}_{||}$
is the electric field component parallel to the interface and
$\mathbf{D}_{\perp}$ is the electric displacement component perpendicular to
the interface. We note that this effect can cause a frequency shift with a
sign different from photoelasticity, and thus can sometimes reduce the single
photon coupling strength.

###  IV.3 Estimation of the single photon coupling strength

The mechanical behaviour of the undercut SiO2 HBAR resonator was simulated via
finite element methods using the COMSOL software, including the piezoelectric
effect. The simulations were used to obtain the admittance curve, from which
the resonances could be determined, and to compute the single photon
optomechanical coupling strength, including both photoelasticity and moving
boundaries effects described in the previous subsections. As the optical
waveguides are made of silicon nitride and silicon oxide in the amorphous
phase, symmetry arguments ensure that the form of the photoelastic
coefficients tensor is of the kind

$\mathbf{p}_{\text{amorphous}}=\left(\begin{matrix}p_{11}&p_{12}&p_{12}&0&0&0\\\
p_{12}&p_{11}&p_{12}&0&0&0\\\ p_{12}&p_{12}&p_{11}&0&0&0\\\
0&0&0&p_{44}&0&0\\\ 0&0&0&0&p_{44}&0\\\ 0&0&0&0&0&p_{44}\end{matrix}\right)$
(39)

with $p_{44}=\frac{1}{2}(p_{11}-p_{12})$. The values of the photoelastic
coefficients of silicon nitride have only been reported by few authors. Here
we chose the latest values reported in literature: $p_{11}=0.239$ [50] and
$p_{12}=0.047$ [51], giving $p_{44}=0.096$. The coefficients used for the
silicon oxide cladding, which contains around $30\%$ of the TE optical mode
volume were: $p_{11}=0.121$, $p_{12}=0.270$ and $p_{44}=-0.0745$ [52].

The inset in Fig. 2a) shows the profile of the norm of the electric field of
the optical mode on which the device will operate, while the core of Fig. 2a)
shows the pattern of the total stress along the vertical direction of an HBAR
resonance. As the acoustic impedance of the Si3N4 waveguide did not match the
acoustic impedance of the rest of the stack, its position had to be optimized
to obtain high optomechanical coupling simultaneously with high
electromechanical coupling. In the end the position of the waveguide could be
chosen such that the strong acoustic resonance around $3.285$GHz shows a
single photon optomechanical coupling rate $\frac{g_{0}}{2\pi}\approx 400$Hz.

##  V Microwave-to-optical transduction

Figure 3:  Microwave-optical conversion efficiency. (a) : Signal flow chart
describing the MW-optical conversion process. The squares represent the input
and output modes, while the circles represent the internal modes of the
converter. The arrows represent the coupling between the modes and are
weighted by the value of the coupling rate between the modes. The transfer
functions between two different modes can be computed with this diagram using
Mason’s gain formula. (b) : Theoretical conversion efficiency from the
microwave port to the transmitted optical asymmetric supermode as a function
of the input pump laser power. The parameters used to draw this curve are
indicated in the textbox as well as in table (1), and correspond to the dashed
red line in Fig. 3b). (c) : Microwave to optical conversion efficiency as a
function of the input pump laser power and the coupling rate from the bus
waveguide to the first ring cavity. The parameters used to draw this curve are
indicated in table (1).

In this section, we finally derive the expression of the conversion efficiency
between the input MW photon and the output upconverted optical photon. Details
on the derivation can be found in the appendices (B) and (C).

Here we consider a pair of coupled optical modes, of which only one is
accessed by a bus waveguide to receive and transmit power, while the other
interacts with the mechanical mode, accessed by a microwave mode. This device
is depicted on Fig. 1. The corresponding Hamiltonian is

$\begin{split}\hat{H}=\
&\hbar\omega_{1}\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hbar\omega_{2}\hat{a}_{2}^{\dagger}\hat{a}_{2}+\hbar\omega_{MW}\hat{b}^{\dagger}\hat{b}\\\
&-\hbar g_{0}\hat{a}_{2}^{\dagger}\hat{a}_{2}(\hat{b}+\hat{b}^{\dagger})-\hbar
J\left(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1}\right)\end{split}$
(40)

The first three terms give the evolution of the subsystems when they are
isolated, the fourth term creates the interaction between the optical mode in
the second ring and the HBAR mechanical mode, and the last term arises from
the coupling between the optical modes of the two microrings. The Hamiltonian
is then transformed following the procedure described in the appendix (B) and
give Langevin equations of the internal modes of the converter as
555$\Delta_{1}=\omega_{L}-\omega_{1}$ and $\Delta_{2}=\omega_{L}-\omega_{2}$
are the detuning of the pump laser to the first and second ring cavities as
defined in section (B.2). In all generality they do not need to have the same
value, even though it was assumed to be the case in section (VI).

(41c)

Combining these with the input-relation for the optical transmission through
the bus waveguide to the first microring cavity

$\hat{a}_{\text{out}}=\hat{a}_{\text{in}}-\sqrt{\kappa_{\text{ex}}}\hat{a}_{1}$
(41ap)

and the microwave reflection from the HBAR

$\hat{c}_{\text{out}}=-\hat{c}_{\text{in}}+\sqrt{\gamma_{\text{ex}}}\hat{b}$
(41aq)

we solved these linear differential equations in the frequency domain,
neglecting the counter-rotating terms in a first time, to obtain the transfer
functions between the input and output modes. Noticing that a few groups of
terms appear recurrently in the transfer functions, it is convenient to define
the susceptibilities of the optical modes and the mechanical mode

$\displaystyle\begin{cases}\chi_{\text{opt},1}[s]&=\frac{1}{s-i\Delta_{1}+\frac{\kappa_{1}}{2}}\\\
\chi_{\text{opt},2}[s]&=\frac{1}{s-i\Delta_{2}+\frac{\kappa_{2}}{2}}\\\
\chi_{m}[s]&=\frac{1}{s+i\omega_{m}+\frac{\gamma_{m}}{2}}\end{cases}$ (41ar)

as well as frequency dependent cooperativities for the optomechanical coupling
and for the coupling between the two rings

$\displaystyle\begin{cases}C_{\text{OM}}[s]&=(\chi_{\text{opt},2}[s]\chi_{m}[s])*g_{\text{OM}}^{2}[s]\\\
C_{\text{OO}}[s]&=\chi_{\text{opt},1}[s]\chi_{\text{opt},2}[s]J^{2}\end{cases}$
(41as)

where $g_{\text{OM}}[s]=g_{0}\bar{a}_{2}[s]$. This contrasts with the usual
definition in the optomechanics community where the cooperativity is a
constant quantity, but has the advantage of taking into account the frequency
dependent behaviour of the total system, and showing that these terms can be
understood as coming from the feedback loops between the different modes. By
further defining frequency dependent extraction efficiencies as

$\displaystyle\begin{cases}\eta_{\text{opt}}[s]&=\frac{\kappa_{\text{ex}}}{2}\chi_{\text{opt},1}[s]\\\
\eta_{\text{MW}}[s]&=\frac{\gamma_{\text{ex}}}{2}\chi_{m}[s]\end{cases}$
(41at)

and choosing to place the pump laser on the symmetric supermodes of the
coupled rings, the conversion efficiency from the microwave input to the
optical output now takes the simple form

$\eta_{\delta\hat{a}_{\text{out}}\hat{c}_{\text{in}}}[\omega]=\frac{4|C_{\text{OM}}[\omega]C_{\text{OO}}[\omega]\eta_{\text{MW}}[\omega]\eta_{\text{opt}}[\omega]|}{|C_{\text{OM}}[\omega]+C_{\text{OO}}[\omega]+1|^{2}}$
(41au)

Fig. 3b) and Fig. 3c) show that, for realistic values of the parameters given
in the table (1), conversion efficiencies above $50\%$, that are limited by
the extraction efficiencies and not the internal conversion process, should
already be within reach of today’s technological capabilities using this new
piezooptomechanical scheme.

We note that the form of the formula for the conversion efficiency is slightly
different from the one reported in literature for transducers relying on only
one optical mode and one microwave mode. The additional terms can be
understood as arising from the feedback loops created by the bidirectional
couplings between the modes. This also implies that this linearized system of
equations can be solved from the diagram (3) by using Mason’s gain formula
[53, 48, 54], similarly to the method introduced by L.Ranzani et al. to study
nonreciprocity [55]. The complete conversion efficiency accounting for the
counter-rotating terms too is the square modulus of the transfer function
$G_{12}[\omega]=\frac{N}{D}$ between $\hat{c}_{\text{in}}$ and
$\hat{a}_{\text{out}}$, where the numerator is

$N=\
\sqrt{\gamma_{\text{ex}}}\sqrt{\kappa_{\text{ex}}}\chi_{1}[\omega]\chi_{2}[\omega]\chi_{m}[\omega]J*g_{\text{OM}}\left(1+\chi_{1}^{*}[\omega]\chi_{2}^{*}[\omega]J^{2}+\chi_{2}^{*}[\omega]\chi_{m}^{*}[\omega]|*g_{\text{OM}}|^{2}\right)$
(41av)

and the denominator is

$\begin{split}D=\
&1+(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})+(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})\\\
&-(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})-(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})\\\
&+(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})+(\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})\\\
&+(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})-(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})\\\
&-(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})\end{split}$
(41aw)

This formula is obtained from Fig. 3a) by applying Mason’s gain rule [53]. The
details of the derivation are reported in the appendix (C.3).

Table 1: Values of the parameters used to compute the conversion efficiencies. Optical frequency | $\frac{\omega_{c}}{2\pi}$ | 193 THz
---|---|---
Optical coupling rate | $\frac{\kappa_{\text{ex}}}{2\pi}$ | 125 MHz
Intrinsic optical linewidth | $\frac{\kappa_{0}}{2\pi}$ | 25.0 MHz
Intrinsic optical quality factor | $Q_{o}$ | $7.5\times 10^{6}$
Single photon optomechanical coupling rate | $\frac{g_{0,\text{OM}}}{2\pi}$ | 400 Hz
Input laser power | $P_{\text{in}}$ | 100 mW
Number of optical photons | $|\bar{a}_{2}|^{2}$ | $\mathcal{O}(10^{8})$
Optomechanical coupling rate | $\frac{g_{\text{OM}}}{2\pi}$ | 20 MHz
Detuning to the symmetric supermode | $\frac{\Delta_{\text{S}}}{2\pi}$ | 0 GHz
Detuning to the asymmetric supermode | $\frac{\Delta_{\text{A}}}{2\pi}$ | 3.267 GHz
Mechanical frequency | $\frac{\omega_{m}}{2\pi}$ | 3.267 GHz
Intrinsic mechanical linewidth | $\frac{\gamma_{0}}{2\pi}$ | 5.3 MHz
Mechanical quality factor | $Q_{m}$ | 600
Electromechanical coupling factor | $k_{\text{eff}}^{2}$ | $4.3\times 10^{-3}$
Figure of merit for piezoelectrics 666Electrical engineers use $M=\frac{k_{\text{eff}}^{2}Q}{1-k_{\text{eff}}^{2}}$ as figure of merit for piezoelectric devices. | $M$ | $2.61$
Electromechanical coupling rate | $\frac{g_{\text{EM}}}{2\pi}$ | 100 MHz
Mechanical coupling rate | $\frac{|\gamma_{\text{ex}}|}{2\pi}$ | 2.9 MHz
Total mechanical linewidth | $\frac{\gamma_{\text{total}}}{2\pi}$ | 8.2 MHz

##  VI Device implementation

Table 2: Comparison of state-of-the-art microwave-to-optical transducers. Reference | Device type | Input power | Cooperativity | Internal efficiency | On-chip efficiency | Temperature | Laser scheme | Waveguide material
---|---|---|---|---|---|---|---|---
[10] | OM | - | - | - | $8.6\times 10^{-2}$ | 4K | CW | Free space
[13] | OM | - | - | - | $4.7\times 10^{-1}$ | 35 mK | CW | Free space
[15] | OM | 625 pW | - | 0.967 | $1.2\times 10^{-2}$ | 50 mK | CW |  Si
[22] | EO | 6 mW | 0.075 | 0.26 | $2.05\times 10^{-2}$ | 1.7 K | CW |  AlN
[23] | EO | 72 $\mu$W | - | 0.0012 | $2.4\times 10^{-5}$ | 40 mK | Pulsed |  AlN
[19] | EO | 20 mW | 0.041 | 0.152 | $1.02\times 10^{-2}$ | 1.9 K | Pulsed |  LiNbO3
[21] | EO | 0.42 mW | 0.004 | - | $1.09\times 10^{-3}$ | RT | CW |  LiNbO3
[16] | EO | - | - | - | $2.7\times 10^{-5}$ | 1 K | CW |  LiNbO3
[30] | PiezoOM | 107 nW | 1.7 | - | $<<2.5\times 10^{-5}$ | 20 mK | Pulsed |  GaAs
[32] | PiezoOM | 20mW | - | - | $8.8\times 10^{-6}$ | 15mK | Pulsed |  Si
[21] | PiezoOM | 3.3 $\mu$W | - | - | $1\times 10^{-5}$ | RT | CW |  LiNbO3
[35] | PiezoOM | 372 $\mu$W | 0.002 | - | $1.2\times 10^{-6}$ | RT | CW |  LiNbO3
This work | PiezoOM | 100 mW | 0.05 |  $>$ 0.9 | $5.0\times 10^{-1}$ | - | CW |  Si3N4

Finally, we give an estimation of realistic values of the parameters to assess
the feasibility of this approach. The values used in this section were
obtained from finite elements simulations, and confirmed with preliminary
experimental data on devices similar to the one proposed here for
transduction.

Fig. 3c) shows the conversion efficiency with the input laser power for
different coupling rate of the bus to the ring cavities. It shows that the
maximal conversion efficiency achievable is increasing with the bus-ring
coupling rate, but the required input power increases similarly. The
requirement of not exceeding the cooling power of the fridge for cryogenic
operations provides a solution to this trade-off.

Fig. 3b) shows the cross-section of Fig. 3c) for a bus-ring coupling rate
$\frac{\kappa_{\text{ex}}}{2\pi}=125$MHz. This value can easily be reached on
devices fabricated with current technologies and enables to achieve $50\%$
conversion efficiency between the microwave and optical domains with a
reasonable input laser power on the symmetric supermode of $100$mW.

The table 2 shows that the approach proposed here can compete the state-of-
the-art of microwave-to-optical transducers. However, this is under the
reserve that no unforeseen phenomenon hinders the conversion process, as it
was the case, for example, for lithium niobate based converters with quasi-
particles formation and photorefractive effect [16, 18, 19]. While the
theoretical results presented here were obtained in the assumption of a
continuous wave pump laser, the same conclusions would applied in the case of
a pulsed pump, for a repetition rate up to the kHz order. The option of a
pulsed operation remains thus available to reach higher intracavity photon
numbers without exceeding the thermal budget of the fridge in which the device
would operate for transduction at quantum levels. We stress that the
conversion efficiency predicted here is already sufficient to perform
experiments in the quantum regime, as the efficiency reaches values above
$50\%$ for reasonably high input pump powers of $\mathcal{O}(100\text{mW})$.
We deduce that the efficiency is not limited by the internal conversion
process, and can thus be improved by overcoupling it. This can be done by
optimizing the position of the bus waveguide relatively to the coupled rings,
or optimizing the geometry of the HBAR resonator to increase the
electromechanical coupling factor. Using another piezoelectric material with a
better electromechanical coupling factor $K^{2}$ is another viable option to
reach a higher $k_{\text{eff}}^{2}$.

##  VII Conclusion

With this paper we propose a new type of transducer between the microwave and
optical domains, harnessing the synergia between photonic integrated circuit
and MEMS technology. The result of the theoretical analysis show that
conversion efficiencies above $50\%$, the threshold required to use quantum
error correction techniques that would enable the conversion of microwave
qubit in the optical domain, is within reach of the current technology [23].
We furthermore tried to clarify the approach that has to be taken when
assessing the performances of devices including several quantum modes.

###### Acknowledgements.

The authors are thankful to Liu Qiu for fruitful discussions on the theory of
optomechanics and Anat Siddharth for insights about experimental aspects of
the measurements of HBAR devices. This material is based upon work supported
by the Air Force Office of Scientific Research under number FA9550-21-1-0047
(Quantum Accelerator). This work was further supported by funding from the
European Union H2020 research and innovation programme under grant agreement
No. 732894 (FET-Proactive HOT).

##  Appendix A Lagrangian formulation of the piezoelectric fields

The derivation of the electromechanical coupling rate from the piezoelectric
interaction requires to quantize the mechanical and electromagnetic modes
inside the piezoelectric material. This can be done easily after obtaining the
proper Hamiltonian densities. To derive the Hamiltonian of the fields it is
useful to first go through their Lagrangian formulation.

The Lagrangian enables to describe the evolution of a system for a given set
of generalized coordinates and their derivatives $\\{\phi_{i},\phi_{i,j}\\}$

$L(\phi_{i},\phi_{i,j})=T-V=\iiint\mathcal{L}(\phi_{i},\phi_{i,j})\
d\mathcal{V}$ (41cd)

thanks to Lagrange’s equations

$\frac{\partial\mathcal{L}(\phi_{i},\phi_{i,j})}{\partial\phi_{i}}=\sum_{j}\frac{\partial}{\partial
x_{j}}\left(\frac{\partial\mathcal{L}(\phi_{i},\phi_{i,j})}{\partial\phi_{i,j}}\right)$
(41ce)

Lagrangian density of the elastic waves is

$\mathcal{L}_{\text{e.w.}}=\sum_{i}\frac{1}{2}\rho\dot{u}_{i}\dot{u}_{i}-\sum_{ijkl}\frac{1}{2}c_{ijkl}u_{i,j}u_{k,l}$
(41cf)

for which the corresponding Hamiltonian density

$\mathcal{H}_{\text{e.w.}}=\frac{1}{2}\rho\dot{u}_{i}\dot{u}_{i}+\frac{1}{2}c_{ijkl}u_{i,j}u_{k,l}$
(41cg)

is derived with Legendre’s transformation
$\mathcal{H}=\sum_{i}\dot{\phi}_{i}\frac{\partial\mathcal{L}}{\partial\dot{\phi}_{i}}-\mathcal{L}$
[56]. By the same principle, the Lagrangian density of the electromagnetic
waves

$\mathcal{L}_{\text{EM}}=-\rho_{e}\phi_{e}+\mathbf{j}\cdot\mathbf{A}+\frac{\epsilon_{0}\mathbf{E}^{2}}{2}-\frac{\mathbf{B}^{2}}{2\mu_{0}}$
(41ch)

leads to the Hamiltonian density [56]

$\mathcal{H}_{\text{EM}}=\frac{1}{2}\left(\mathbf{E}\cdot\mathbf{D}+\mathbf{H}\cdot\mathbf{B}\right)$
(41ci)

##  Appendix B Transformation of the Hamiltonian

###  B.1 Unitary transformation

In this appendix, we justify the formula used to apply a unitary
transformation to the Hamiltonian.

Starting from Schrodinger equation

$i\hbar\frac{\partial}{\partial t}\ket{\psi}=\hat{H}\ket{\psi}$ (41cj)

and the new state $\ket{\tilde{\psi}}=\hat{U}\ket{\psi}$ obtained after
applying the unitary transformation $\hat{U}$, we obtain

$\begin{split}&i\hbar\frac{\partial}{\partial
t}\left(\hat{U}^{\dagger}\ket{\tilde{\psi}}\right)=\hat{H}\hat{U}^{\dagger}\ket{\tilde{\psi}}\\\
\Leftrightarrow\quad&i\hbar\frac{\partial}{\partial
t}\left(\hat{U}^{\dagger}\right)\ket{\tilde{\psi}}+i\hbar\hat{U}^{\dagger}\frac{\partial}{\partial
t}\left(\ket{\tilde{\psi}}\right)=\hat{H}\hat{U}^{\dagger}\ket{\tilde{\psi}}\\\
\Leftrightarrow\quad&i\hbar\hat{U}^{\dagger}\frac{\partial}{\partial
t}\left(\ket{\tilde{\psi}}\right)=\hat{H}\hat{U}^{\dagger}\ket{\tilde{\psi}}-i\hbar\frac{\partial}{\partial
t}\left(\hat{U}^{\dagger}\right)\ket{\tilde{\psi}}\\\
\Leftrightarrow\quad&i\hbar\frac{\partial}{\partial
t}\left(\ket{\tilde{\psi}}\right)=\hat{U}\hat{H}\hat{U}^{\dagger}\ket{\tilde{\psi}}-i\hbar\hat{U}\frac{\partial}{\partial
t}\left(\hat{U}^{\dagger}\right)\ket{\tilde{\psi}}\end{split}$ (41ck)

which justifies the form of the Hamiltonian in the new frame

$\displaystyle\hat{H}^{\text{new}}=\hat{U}\hat{H}^{\text{old}}\hat{U}^{\dagger}-i\hbar\hat{U}\frac{\partial\hat{U}^{\dagger}}{\partial
t}$ (41cl)

&

###  B.2 Rotating frame

The form of the Hamiltonian in the frame rotating at the frequency of the pump
laser $\omega_{L}$ is computed using the relation derived in the previous
section.

In this case the transformation is given by
$\hat{U}=e^{i\hat{a}^{\dagger}\hat{a}\omega_{L}t}$ so that
$\ket{\psi^{\text{old}}}=e^{-i\hat{a}^{\dagger}\hat{a}\omega_{L}t}\ket{\psi^{\text{new}}}$,
i.e. the optical wavefunctions in the new frame correspond to slowly varying
envelopes. Using the fact that

$\hat{U}(\hbar\omega_{c}\hat{a}^{\dagger}\hat{a})\hat{U}^{\dagger}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}$
(41cm)

and

$\frac{\partial\hat{U}^{\dagger}}{\partial
t}=-i\omega_{L}\hat{a}^{\dagger}\hat{a}\ \hat{U}^{\dagger}$ (41cn)

the effect of applying
$\hat{U}=e^{i\omega_{L}(\hat{a}_{1}^{\dagger}\hat{a}_{1}+\hat{a}_{2}^{\dagger}\hat{a}_{2})t}$
to move in the rotating frame is only to transform the terms from the isolated
optical subsystems into

$\hat{H}^{\text{new}}=\hbar\omega_{c}\hat{a}^{\dagger}\hat{a}-\hbar\omega_{L}\hat{a}^{\dagger}\hat{a}=-\hbar\Delta\hat{a}^{\dagger}\hat{a}$
(41co)

where we defined the detuning of the optical mode to the pump laser as
$\Delta=\omega_{L}-\omega_{c}$.

###  B.3 Optomechanical interaction linearization

The optomechanical interaction Hamiltonian [6]

$\hat{H}_{\text{OM}}=-\hbar
g_{0}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})$ (41cp)

reveals itself to be problematic when looking at the evolution of annihilation
operators because of the triple product, that leaves nonlinear terms in the
Langevin equations. Conventional solution to this problem used by the
community of researchers in the field of optomechanics is to linearize the
optical field amplitude around the average value of the field in the cavity
$\bar{a}=\expectationvalue{\hat{a}}$. This approximation consists in
separating the mean value and the fluctuations, which take into account for
the quantum properties we are interested in with this transduction scheme
proposal,

$\hat{a}\approx\bar{a}+\delta\hat{a}$ (41cq)

With this expansion of the optical field, the optomechanical Hamiltonian
becomes

$\begin{split}&\hbar
g_{0}\hat{a}^{\dagger}\hat{a}(\hat{b}+\hat{b}^{\dagger})\approx\\\ &\hbar
g_{0}\left[|\bar{a}|^{2}+\delta\hat{a}^{\dagger}\delta\hat{a}+\bar{a}\delta\hat{a}^{\dagger}+\bar{a}^{*}\delta\hat{a}\right](\hat{b}+\hat{b}^{\dagger})\end{split}$
(41cr)

where we can identify three kind of terms:

  1. 1.

a constant term causing a shift of the displacement origin $\hbar
g_{0}|\bar{a}|^{2}(\hat{b}+\hat{b}^{\dagger})$

  2. 2.

a term that will be kept as the linearized optomechanical Hamiltonian $\hbar
g_{0}\left[\bar{a}\delta\hat{a}^{\dagger}+\bar{a}^{*}\delta\hat{a}\right](\hat{b}+\hat{b}^{\dagger})$

  3. 3.

and a term $\hbar
g_{0}\delta\hat{a}^{\dagger}\delta\hat{a}(\hat{b}+\hat{b}^{\dagger})$ that is
neglected for having a contribution at least $|\bar{a}|$ smaller than the two
other, which rapidly becomes consequent for reasonable pump power.

The first disappears by applying an appropriate shift $\delta x$ of the
displacement origin. The value of this shift is obtained by evaluating the
susceptibility of the mechanical oscillator in the steady state
$\chi[0]=\frac{1}{k}=\frac{1}{m_{\text{eff}}\omega_{m}^{2}}$ multiplied by the
average radiation pressure force $\bar{F}_{\text{rad. press.}}=\hbar
G|\bar{a}|^{2}$, giving the displacement at equilibrium

$\delta x=\frac{\hbar G|\bar{a}|^{2}}{m_{\text{eff}}\omega_{m}^{2}}$ (41cs)

for a frequency shift per displacement unit
$G=-\frac{\partial\omega_{\text{cav}}}{\partial
x}=\frac{g_{0}}{x_{\text{ZPF}}}$. Writing the displacement and the associated
momentum in terms of annihilation operators

$\displaystyle\begin{cases}\hat{x}=\sqrt{\frac{\hbar}{2m_{\text{eff}}\omega_{m}}}(\hat{b}+\hat{b}^{\dagger})\\\
\hat{p}_{x}=\sqrt{\frac{\hbar
m_{\text{eff}}\omega_{m}}{2}}\frac{1}{i}(\hat{b}-\hat{b}^{\dagger})\end{cases}$
(41ct)

we can then easily construct the appropriate translation operator as [57]

$\hat{T}(\delta x)=e^{\frac{1}{i\hbar}\delta x\hat{p}_{x}}$ (41cu)

Applying this operator to the original Hamiltonian, we find that the first
term occurring in the linearization of the optomechanical Hamiltonian vanishes
at the cost of shifting the optical resonances frequency, leading to a
modified detuning

$\Delta_{\text{new}}=\Delta_{\text{old}}+G\delta\bar{x}$ (41cv)

###  B.4 Intracavity fields

Figure 4:  Optical transmission of the coupled microrings. a) : Signal flow
chart describing the behaviour of the two coupled optical modes, without the
effect of the mechanics. The transfer functions of the intracavity fields can
also be obtained by applying Mason’s gain rule to this diagram. b) : Optical
transmission of the coupled rings system. Both rings are set to have the same
frequency $\omega_{c}$ (there is on detuning $\delta$ between the rings
resonant frequency). The strong coupling $J$ between them causes modes
hybridization, which results in two effective resonance frequency separated by
$2J$. The two absorption peaks corresponds to the symmetric (lower frequency)
and asymmetric supermodes of the photonic dimer. The transmission spectrum is
showed in the frame rotating at $\omega_{c}$.

In this section we explain how to obtain the intracavity photon numbers for
the two rings of the photonic dimer, terms that arise in the optomechanical
coupling rate.

We consider a pair of coupled optical modes from which only the first one can
be probed via a bus waveguide. This situation is illustrated with the signal
flow graph on Fig. 4. Linearizing the annihilation operators of the optical
modes as described in the previous section, we can separate the evolution of
the fluctuations from the evolution of the average cavity field

$\displaystyle\frac{d}{dt}\hat{a}_{1}=\frac{d}{dt}\bar{a}_{1}+\frac{d}{dt}\delta\hat{a}_{1}$
(41cw)
$\displaystyle\frac{d}{dt}\hat{a}_{2}=\frac{d}{dt}\bar{a}_{2}+\frac{d}{dt}\delta\hat{a}_{2}$
(41cx)

We group the terms with $\delta\hat{a}_{1}$ and $\delta\hat{a}_{2}$ to obtain
the Langevin equations for the fluctuations operators

$\displaystyle\frac{d}{dt}\delta\hat{a}_{1}=\ $ $\displaystyle
i\Delta_{1}\delta\hat{a}_{1}-\frac{\kappa_{1}}{2}\delta\hat{a}_{1}+iJ\delta\hat{a}_{2}$
$\displaystyle+\sqrt{\kappa_{0,1}}\hat{f}_{o,1}+\sqrt{\kappa_{\text{ex},1}}\delta\hat{a}_{\text{in}}$
(41cy) $\displaystyle\frac{d}{dt}\delta\hat{a}_{2}=\ $ $\displaystyle
i\Delta_{2}\delta\hat{a}_{2}-\frac{\kappa_{2}}{2}\delta\hat{a}_{2}+iJ\delta\hat{a}_{1}$
$\displaystyle+ig_{0}\bar{a}_{2}(\hat{b}+\hat{b}^{\dagger})+\sqrt{\kappa_{0,2}}\hat{f}_{o,2}$
(41cz)

while the remaining terms give the equation of motion of the average
intracavity fields

$\displaystyle\frac{d}{dt}\bar{a}_{1}$
$\displaystyle=-i\omega_{1}\bar{a}_{1}-\frac{\kappa_{1}}{2}\bar{a}_{1}+iJ\bar{a}_{2}+\sqrt{\kappa_{\text{ex},1}}\bar{a}_{\text{in}}$
(41da) $\displaystyle\frac{d}{dt}\bar{a}_{2}$
$\displaystyle=-i\omega_{2}\bar{a}_{2}-\frac{\kappa_{2}}{2}\bar{a}_{2}+iJ\bar{a}_{1}$
(41db)

Taking Laplace’s transform, we then obtain the values of the intracavity
fields in the frequency domain

$\displaystyle\begin{cases}\bar{a}_{1}[s]&=\sqrt{\kappa_{\text{ex}}}\frac{\chi_{1}[s]}{\left(1+J^{2}\chi_{2}[s]\chi_{1}[s]\right)}\bar{a}_{\text{in}}[s]\\\
\bar{a}_{2}[s]&=iJ\chi_{2}[s]\bar{a}_{1}[s]\end{cases}$ (41dc)

The output spectrum corresponding to these intracavity fields is shown on Fig.
4.

###  B.5 Input laser spectrum

We assume an input laser with a Lorentzian lane shape, with linewidth $k_{L}$
and total power $P_{\text{in}}$.

Setting the laser spectrum to be

$\bar{a}_{\text{in}}[\omega]=\frac{1}{\sqrt{\pi}}\frac{\sqrt{\frac{\kappa_{L}}{2}}}{-i(\omega-\omega_{L})+\frac{\kappa_{L}}{2}}\sqrt{\frac{P_{in}}{\hbar\omega_{L}}}e^{i\phi_{L}}$
(41dd)

it possesses the correct linewidth and gives the correct total number of
photons

$\int_{-\infty}^{\infty}|\bar{a}_{\text{in}}[\omega]|^{2}d\omega=\frac{P_{\text{in}}}{\hbar\omega_{L}}$
(41de)

Taking, for example, a total power of $10$mW and a linewidth of
$\frac{\kappa_{L}}{2\pi}=10$kHz for the pump laser, which are realistic
values, its spectrum is very narrow compared to all the other MHz order
linewidth of the modes in the transducer. This justifies that we assume a
Dirac peak for the laser spectrum during the computation of the conversion
efficiency.

###  B.6 On the importance of pumping on the optical supermodes

For strong ring-ring coupling $J$, coupled rings cavities undergo mode
hybridization. In the main text we assumed that the pump laser is centered on
the symmetric supermode, which has a lower frequency than the asymetric one.
While pumping on the symmetric supermode is important to cool the mechanical
mode, this assumption is also important for a simplification it brings of the
literal treatment of the equations.

The treatment showed in the main text shows the mean field in the second
cavity $\bar{a}_{2}$ as being a constant. This is only valid if the system is
pumped on one of its supermodes with a continuous wave laser, which is what we
assumed here. If we were to consider different conditions, it would be
necessary to reintroduce the temporal behaviour of $\bar{a}_{2}(t)$. This is
problematic because the Langevin equations of the second ring optical
fluctuations and mechanical mode contain terms
$ig_{0}\bar{a}_{2}(t)\hat{b}(t)$ and
$ig_{0}\bar{a}_{2}^{*}(t)\delta\hat{a}_{2}(t)$ in the temporal domain, which
become convolutional products
$ig_{0}\frac{1}{2\pi}\int\bar{a}_{2}[\omega-\sigma]\hat{b}[\sigma]\ d\sigma$
and
$ig_{0}\frac{1}{2\pi}\int\bar{a}_{2}^{*}[\omega-\sigma]\delta\hat{a}_{2}[\sigma]\
d\sigma$ in the frequency domain. Pumping on the symmetric supermode with a
laser spectrum having a Dirac distribution (as justified in the subsection
(B.5)), the spectrum of the mean cavity field of the second ring becomes
$\bar{a}_{2}[\omega]=A_{2}\delta[\omega+\Delta_{S}]$ with $A_{2}$ the
amplitude determined by the pump power as indicated in the subsection (B.4).
Pumping on the asymmetric supermode instead leads to the same kind of
treatment. This spectrum then leads to terms
$ig_{0}\frac{1}{2\pi}A_{2}\hat{b}[\omega+\Delta_{S}]$ and
$ig_{0}\frac{1}{2\pi}A_{2}\delta\hat{a}_{2}[\omega-\Delta_{S}]$. That enables
us to apply linear system theory as long as we take into account the frequency
shifts. We note that these frequency shifts are responsible for the conversion
of photons between the optical modes, which is an essential characteristic of
the converter.

##  Appendix C Transfer functions

In this appendix we give a brief reminder on notions of control theory applied
here to solve the linear system of equations describing the transducer. The
complete set of transfer functions is given applying this state space model,
while neglecting the counter-rotating terms The derivation of the microwave-
optical transfer function with counter-rotating terms is obtained using the
signal flow graph 3 and Mason’s gain rule.

###  C.1 State space representation

The Langevin equation obtained by correspondance rules from Poisson’s bracket
[57]

$\frac{d}{dt}\hat{a}=\left[\hat{a},\hat{H}\right]+\frac{\partial\hat{a}}{\partial
t}$ (41df)

is applied to the Hamiltonian (40) to obtain the equation of motion of the
internal modes of the transducer. Neglecting the counter-rotating terms that
are not playing a significant role here, the Langevin equations describing the
conversion process of the transducer are

$\displaystyle\frac{d}{dt}\delta\hat{a}_{1}$
$\displaystyle=i\Delta_{1}\delta\hat{a}_{1}-\frac{\kappa_{1}}{2}\delta\hat{a}_{1}+iJ\delta\hat{a}_{2}$
$\displaystyle+\sqrt{\kappa_{0,1}}\hat{f}_{o,1}+\sqrt{\kappa_{\text{ex},1}}\delta\hat{a}_{\text{in}}$
(41dg) $\displaystyle\frac{d}{dt}\delta\hat{a}_{2}$
$\displaystyle=i\Delta_{2}\delta\hat{a}_{2}-\frac{\kappa_{2}}{2}\delta\hat{a}_{2}$
$\displaystyle+iJ\delta\hat{a}_{1}+ig_{0}\bar{a}_{2}\hat{b}+\sqrt{\kappa_{0,2}}\hat{f}_{o,2}$
(41dh) $\displaystyle\frac{d}{dt}\hat{b}$
$\displaystyle=-i\omega_{m}\hat{b}-\frac{\gamma_{m}}{2}\hat{b}+ig_{0}\bar{a}_{2}^{*}\delta\hat{a}_{2}$
$\displaystyle+\sqrt{\gamma_{0}}\hat{f}_{m}+\sqrt{\gamma_{\text{ex}}}\hat{c}_{\text{in}}$
(41di)

It is convenient to write this linear system of differential equations in a
matrix form

$\dot{\mathbf{x}}=\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$ (41dj)

with the state vector

$\mathbf{x}=\begin{pmatrix}\delta\hat{a}_{1}\\\ \delta\hat{a}_{2}\\\
\hat{b}\end{pmatrix}$ (41dk)

and the input vector

$\mathbf{u}=\begin{pmatrix}\hat{a}_{\text{in}}\\\ \hat{c}_{\text{in}}\\\
\hat{f}_{o,1}\\\ \hat{f}_{o,2}\\\ \hat{f}_{m}\end{pmatrix}$ (41dl)

where the first two terms are the input fields in the device, and the
remaining three comes from the noises. The system matrix is then given by

$\mathbf{A}=\begin{pmatrix}i\Delta_{1}-\frac{\kappa_{1}}{2}&iJ&0\\\ \\\
iJ&i\Delta_{2}-\frac{\kappa_{2}}{2}&*ig_{\text{OM}}\\\ \\\
0&*ig_{\text{OM}}&-\frac{\gamma}{2}-i\omega_{m}\end{pmatrix}$ (41dm)

and the input matrix is

$\mathbf{B}=\left(\begin{matrix}\sqrt{\kappa_{\text{ex}}}&0&\sqrt{\kappa_{0,1}}&0&0\\\
\\\ 0&0&0&\sqrt{\kappa_{0,2}}&0\\\ \\\
0&\sqrt{\gamma_{\text{ex}}}&0&0&\sqrt{\gamma_{0}}\end{matrix}\right)$ (41dn)

The input-output relations [8]

$\displaystyle\hat{a}_{\text{out}}=\hat{a}_{\text{in}}-\sqrt{\kappa_{\text{ex}}}\hat{a}_{1}$
(41do)
$\displaystyle\hat{c}_{\text{out}}=-\hat{c}_{\text{in}}+\sqrt{\gamma_{\text{ex}}}\hat{b}$
(41dp)

then give the output matrix

$\mathbf{C}=\begin{pmatrix}-\sqrt{\kappa_{\text{ex}}}&0&0\\\ \\\
0&0&\sqrt{\Gamma_{\text{ex}}}\end{pmatrix}$ (41dq)

and the feedthrough matrix

$\mathbf{D}=\begin{pmatrix}1&0&0&0&0\\\ \\\ 0&-1&0&0&0\end{pmatrix}$ (41dr)

for an output vector

$\mathbf{y}=\begin{pmatrix}\hat{a}_{\text{out}}\\\
\hat{c}_{\text{out}}\end{pmatrix}=\mathbf{C}\mathbf{x}+\mathbf{D}\mathbf{u}$
(41ds)

Taking Laplace’s transformation we have in the frequency domain

$\begin{split}s\tilde{\mathbf{x}}[s]-\mathbf{x}(0)&=\mathbf{A}\tilde{\mathbf{x}}[s]+\mathbf{B}\tilde{\mathbf{u}}[s]\\\
\Leftrightarrow\tilde{\mathbf{x}}[s]&=(s\mathbf{I}-\mathbf{A})^{-1}\left(\mathbf{x}(0)+\mathbf{B}\tilde{\mathbf{u}}[s]\right)\end{split}$
(41dt)

that gives the output in the frequency domain

$\begin{split}\tilde{\mathbf{y}}[s]&=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{x}(0)\\\
&+\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}\tilde{\mathbf{u}}[s]+\mathbf{D}\tilde{\mathbf{u}}[s]\end{split}$
(41du)

Assuming that $\mathbf{x}(0)=\mathbf{0}$, the frequency domain input-output
relation $\tilde{\mathbf{y}}[s]=\mathbf{G}[s]\tilde{\mathbf{u}}[s]$ is
obtained for a transfer functions matrix

$\mathbf{G}[s]=\mathbf{C}(s\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+\mathbf{D}$
(41dv)

###  C.2 Transfer functions of the transducer

The transfer function for the annihilation operators neglecting the counter-
rotating terms are for the optical output $\hat{a}_{\text{out}}$

  * •

from the optical input $\hat{a}_{\text{in}}\rightarrow\hat{a}_{\text{out}}$

$G_{11}=\frac{C_{\text{OM}}+C_{\text{OO}}-2\eta_{\text{opt}}\left(C_{\text{OM}}+1\right)+1}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41dw)

  * •

from the microwave input $\hat{c}_{\text{in}}\rightarrow\hat{a}_{\text{out}}$

$G_{12}=\frac{2\sqrt{C_{\text{OM}}}\sqrt{C_{\text{OO}}}\sqrt{\eta_{\text{MW}}}\sqrt{\eta_{\text{opt}}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41dx)

  * •

from the noise in the first optical ring cavity
$\hat{f}_{o,1}\rightarrow\hat{a}_{\text{out}}$

$G_{13}=-\frac{2\sqrt{\eta_{\text{opt}}}\sqrt{\theta_{o,1}}\left(C_{\text{OM}}+1\right)}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41dy)

  * •

from the noise in the second optical ring cavity
$\hat{f}_{o,2}\rightarrow\hat{a}_{\text{out}}$

$G_{14}=-\frac{2i\sqrt{C_{\text{OO}}}\sqrt{\eta_{\text{opt}}}\sqrt{\theta_{o,2}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41dz)

  * •

from the noise in the mechanical resonator
$\hat{f}_{m}\rightarrow\hat{a}_{\text{out}}$

$G_{15}=\frac{2\sqrt{C_{\text{OM}}}\sqrt{C_{\text{OO}}}\sqrt{\eta_{\text{opt}}}\sqrt{\theta_{m}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41ea)

and for the microwave output $\hat{c}_{\text{out}}$

  * •

from the optical input $\hat{a}_{\text{in}}\rightarrow\hat{c}_{\text{out}}$

$G_{21}=\frac{-2\sqrt{C_{\text{OM}}}\sqrt{C_{\text{OO}}}\sqrt{\eta_{\text{MW}}}\sqrt{\eta_{\text{opt}}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41eb)

  * •

from the microwave input $\hat{c}_{\text{in}}\rightarrow\hat{c}_{\text{out}}$

$G_{22}=\frac{-C_{\text{OM}}-C_{\text{OO}}+2\eta_{\text{MW}}\left(C_{\text{OO}}+1\right)-1}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41ec)

  * •

from the noise in the first optical ring cavity
$\hat{f}_{o,1}\rightarrow\hat{c}_{\text{out}}$

$G_{23}=\frac{-2\sqrt{C_{\text{OM}}}\sqrt{C_{\text{OO}}}\sqrt{\eta_{\text{MW}}}\sqrt{\theta_{o,1}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41ed)

  * •

from the noise in the second optical ring cavity
$\hat{f}_{o,2}\rightarrow\hat{c}_{\text{out}}$

$G_{24}=\frac{2i\sqrt{C_{\text{OM}}}\sqrt{\eta_{\text{MW}}}\sqrt{\theta_{o,2}}}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41ee)

  * •

from the noise in the mechanical resonator
$\hat{f}_{m}\rightarrow\hat{c}_{\text{out}}$

$G_{25}=\frac{2\sqrt{\eta_{\text{MW}}}\sqrt{\theta_{m}}\left(C_{\text{OO}}+1\right)}{C_{\text{OM}}+C_{\text{OO}}+1}$
(41ef)

###  C.3 Mason’s gain of the microwave-to-optical transduction process

The transfer function from the input microwave mode to the output optical mode
can be obtained from Fig. 3 by applying Mason’s rule for the gain of signal
flow graph [53]

$G=\frac{\sum_{k}G_{k}\Delta_{k}}{\Delta}$ (41eg)

where the sum acts on all the possible forward paths from the input mode, the
source, to the output mode, the sink. $G_{k}$ is the gain of the
$k^{\text{th}}$ forward path, $\Delta_{k}$ is the subdeterminant for the loops
not touching the $k^{\text{th}}$ forward path, and
$\Delta=1-\sum_{i}L_{i}+\sum_{i,j}^{\text{not touching}}L_{i}L_{j}+\dots$ is
the total determinant obtained from the individual loop gains $L_{i}$. The
conversion efficiency of the transducer is obtained by first computing the
transfer function $G$ between the annihilation operators $\hat{c}_{\text{in}}$
and $\hat{a}_{\text{out}}$, and then taking the square of its norm $|G|^{2}$.

Here only one path joins $\hat{c}_{\text{in}}$ and $\hat{a}_{\text{out}}$
without going through the same nodes more than once:
$\hat{c}_{\text{in}}\rightarrow\hat{b}\rightarrow\hat{a}_{2}\rightarrow\hat{a}_{1}\rightarrow\hat{a}_{\text{out}}$.
The authors would like to bring the attention of the reader to the fact that
the self-loops of each mode have to be split in order to see the
susceptibilities appearing. The gain of this unique forward path is thus
$\sqrt{\kappa_{\text{ex}}}\sqrt{\gamma_{\text{ex}}}J\chi_{m}[s]\chi_{o,2}[s]\chi_{o,1}[s](*g_{\text{OM}})$.

The next step to compute this transfer function is to identify all the loops
of the graph. The loops with only one node are treated by considering that
they are split in the following. No loop with more than two nodes is present
in this graph. The loops to take into account are thus:

  * •

$\hat{b}\leftrightarrow\hat{a}_{2}:L_{1}=-\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2}$

  * •

$\hat{b}^{\dagger}\leftrightarrow\hat{a}_{2}^{\dagger}:L_{2}=-\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2}$

  * •

$\hat{a}_{1}\leftrightarrow\hat{a}_{2}:L_{3}=-\chi_{o,1}[s]\chi_{o,2}[s]J^{2}$

  * •

$\hat{a}_{1}^{\dagger}\leftrightarrow\hat{a}_{2}^{\dagger}:L_{4}=-\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2}$

  * •

$\hat{b}\leftrightarrow\hat{a}_{2}^{\dagger}:L_{5}=\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2}$

  * •

$\hat{b}^{\dagger}\leftrightarrow\hat{a}_{2}:L_{6}=\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2}$

Only the loops $L_{2}$ and $L_{4}$ do not touch the forward path, and they do
overlap on the node $\hat{a}_{2}^{\dagger}$. The subdeterminant of the forward
path is thus

$\displaystyle\Delta_{\text{forward path}}=1$
$\displaystyle+\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2}$
$\displaystyle+\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2}$ (41eh)

The final step to obtain the total determinant is to identify the loops that
are not touching:

  * •

$L_{1}$ with $\left[L_{2},\ L_{4}\right]$

  * •

$L_{2}$ with $\left[L_{1},\ L_{3}\right]$

  * •

$L_{3}$ with $\left[L_{2},\ L_{4},\ L_{5}\right]$

  * •

$L_{4}$ with $\left[L_{1},\ L_{3},\ L_{6}\right]$

  * •

$L_{5}$ with $\left[L_{3},\ L_{6}\right]$

  * •

$L_{6}$ with $\left[L_{4},\ L_{5}\right]$

The total determinant of the graph is thus

$\begin{split}\Delta_{\text{total}}=&\
1-L_{1}-L_{2}-L_{3}-L_{4}-L_{5}-L_{6}+L_{1}L_{2}+L_{1}L_{4}+L_{2}L_{3}+L_{3}L_{4}+L_{3}L_{5}+L_{4}L_{6}+L_{5}L_{6}\\\
=&\
1+(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})+(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})\\\
&+(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})-(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})-(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})\\\
&-(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(-\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})-(\chi_{o,2}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(-\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})\\\
&-(\chi_{o,2}^{*}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})(-\chi_{o,1}[s]\chi_{o,2}[s]J^{2})-(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})(-\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})\\\
&-(\chi_{o,1}[s]\chi_{o,2}[s]J^{2})(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})-(\chi_{o,1}^{*}[s]\chi_{o,2}^{*}[s]J^{2})(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})\\\
&+(\chi_{o,2}^{*}[s]\chi_{m}[s]|*g_{\text{OM}}|^{2})(\chi_{o,2}[s]\chi_{m}^{*}[s]|*g_{\text{OM}}|^{2})\end{split}$
(41ei)

Mason’s rule (41eg) can now be applied to find the final result of section
(V), equations (41av) and (41aw).

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