of 11
Supplemental Document
Electro-optic transduction in silicon via
gigahertz-frequency nanomechanics:
supplement
H
AN
Z
HAO
,
1,2
A
LKIM
B
OZKURT
,
1,2
AND
M
OHAMMAD
M
IRHOSSEINI
1,2,
1
The Gordon and Betty Moore Laboratory of Engineering, California Institute of Technology, Pasadena,
California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California
91125, USA
mohmir@caltech.com
This supplement published with Optica Publishing Group on 20 June 2023 by The Authors under
the terms of the Creative Commons Attribution 4.0 License in the format provided by the authors
and unedited. Further distribution of this work must maintain attribution to the author(s) and the
published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.23226758
Parent Article DOI: https://doi.org/10.1364/OPTICA.479162
Supplementary Information: Electro-optic transduction in silicon via GHz-frequency
nanomechanics
Han Zhao,
1, 2
Alkim Bozkurt,
1, 2
and Mohammad Mirhosseini
1, 2,
1
The Gordon and Betty Moore Laboratory of Engineering,
California Institute of Technology, Pasadena, California 91125
2
Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, California 91125
(Dated: May 25, 2023)
I. DERIVATION OF THE ELECTROMECHANICAL CONVERSION
We can express the current passing through a motion-dependent capacitor,
C
(
x
), as [1]
i
(
t
) =
C
(
x
)
d
V
d
t
+
d
C
d
x
d
x
d
t
V
=
i
conductive
+
i
motional
.
(1)
As evident, the product of velocity and voltage gives rise to the motional current, which in the most general case
includes multiple frequency components. Assuming a static voltage bias of
V
b
, we find the RF component of the
motional current as
i
motional
,
RF
(
t
) =
V
b
d
C
d
x
d
x
d
t
.
(2)
The rate of energy loss of the mechanical resonator from the motional current can be written as
P
em
=
Z
0
V
b
2

d
C
d
x

2

d
x
d
t

2
,
(3)
where
Z
0
is the impedance of the microwave waveguide. This energy loss rate can be readily converted to an
electromechanical dissipation rate upon division by the total energy,
E
m
, stored in the mechanical oscillator
γ
em
=
P
em
E
m
=
Z
0
V
b
2
m
eff

d
C
d
x

2
.
(4)
Here, we have used
E
m
=
m
eff
d
x
d
t

2
, where
m
eff
is the effective mass of the mechanical resonance.
To determine the electromechanical dissipation rate
γ
em
from simulations, it is necessary to express the change of the
capacitance per displacement
∂C/∂x
for a given set of mechanical, microwave, and electrostatic modes. Mechanical
displacement can create capacitance change via the photoelastic effect, where the stress field alters the permittivity
of silicon. Additionally, we can get a change in the capacitance from the moving-boundary effect, where the material
boundaries deform with the mechanical motion. Denoting the two contribution as
∂C
∂x
ω
m
=
∂C
∂x
PE
+
∂C
∂x
MB
,
(5)
we can write explicit forms for the photoelastic and moving-boundary contributions
∂C
∂x
PE
=
ε
2
ε
0
V
dc
V
rf
ZZZ
V
[
E
dc
·
(
P
·
S
)
·
E
rf
] d
V,
(6)
∂C
∂x
MB
=
1
V
dc
V
rf
ZZ
S
h
(
Q
·
ˆ
n
)(∆
εE
dc
E
rf
ε
1
D
dc
D
rf
)
i
d
S.
(7)
mohmir@caltech.edu; http://qubit.caltech.edu
2
Here,
P
and
S
are the photoelastic and strain tensors,
ε
is the dielectric permittivity of silicon, ∆
ε
=
ε
1
ε
2
and
ε
1
= 1
1
1
2
are the permittivity contrast between the two materials across the boundary. The displacement
field
Q
is normalized such that max(
|
Q
|
) = 1, The quantities
V
dc
,V
rf
denote the voltage difference values across the
capacitor electrodes and are related to the electric fields as
V
dc(rf)
=
Z
E
dc(rf)
·
d
l
,
(8)
where the integral can be taken over any path connecting the two electrodes (we are using the quasi-static approxi-
mation for all the fields such that
E
=
−∇
V).
Due to the non-zero resistivity of the silicon device layer (
3kΩ
.
cm), static fields are expected to be screened by
the free carriers and vanish inside the bulk. To model this effect, we treat silicon as a conductor for simulating the
distribution of the DC biasing field. Considering eq. (6) and eq. (7), this assumption results in the vanishing of the
photoelastic contribution [2], and also a simplification to the moving-boundary component, where only the term with
perpendicular field components remains in place:
∂C
∂x
ω
m
=
1
V
dc
V
rf
ZZ
S
(
Q
·
ˆ
n
)∆
ε
1
D
dc
D
rf
dS
(9)
Plugging eq. (9) into eq. (4), the microwave-to-mechanical external coupling can be expressed as
γ
em
(
V
b
) =
V
2
b
̃
γ
em
.
(10)
Here ̃
γ
em
is the per-volt electromechanical dissipation rate defined as
̃
γ
em
=
Z
0
m
eff
V
dc
2
V
rf
2

ZZ
S
(
Q
·
ˆ
n
)∆
ε
1
D
dc
D
rf
dS

2
.
(11)
So far, we have considered the case of direct coupling to a waveguide. Alternatively, the mechanical resonator
may be coupled to the waveguide via an intermediate microwave resonator. In this situation, the electromechanical
interaction gives rise to a coupling rate between the modes given by [3]
g
em
=
r
Z
r
ω
m
4
m
eff
d
C
d
x
!
V
b
.
(12)
Here,
Z
r
represents the impedance of the microwave resonator. In this situation, the mechanical mode’s decay rate
via the microwave resonator can be found as
γ
em
=
4
g
em
2
κ
mic
=
Z
r
V
b
2
m
eff

d
C
d
x

2

ω
m
κ
mic

,
(13)
where
κ
mic
is the microwave resonator’s linewidth. Comparing this expression with the one in 4, we observe that
using a narrow linewidth and high-impedance microwave resonator can significantly increase the electromechanical
decay rate and, subsequently, the cooperativity.
II. BAND STRUCTURE OF THE PHONON SHIELDS
We use phonon shields with unit cell shown in fig. 1a to clamp the two ends of the nanobeam. The phonon shields
have a mechanical band structure with a compete band gap from 4.3 GHz to 6 GHz (fig. 1b), which terminates the
hybridized mechanical modes at 5 GHz. This design helps the confinement of the mechanical modes within the
nanobeam by preventing the radiative leakage into the surrounding membrane.
III. SIMULATIONS OF THE HYBRIDIZED MECHANICAL MODES WITH AN OFFSET OF THE
EMC SECTION
The frequency conversion efficiency (eq. (23)) is critically dependent on the hybridization of the mechanical modes
that leads to simultaneously large optomechanical coupling and electromechanical conversion. As demonstrated in
3
X
M
0
2
4
6
8
10
F
r
e
q
u
e
n
c
y
(
G
H
z
)
a
b
375 nm
530 nm
76 nm
FIG. 1. Design of the phonon shields. (a) Geometry of the unit cell. (b) Mechanical band structure. The complete band gap
is highlighted by the shaded zone. Red dashed line denotes the frequencies of the 5GHz mechanical modes.
the main text (Fig. 2), the matching of the mechanical resonance frequencies in the EMC and OMC sections, ensures
the formation of the desired supermode. However, in practice, the nanofabrication of the devices can induce disorders
that create a frequency offset between the EMC and OMC mechanical modes. Such disorders may be resulted from
several factors such as non-uniformity of the hole array pattern, thinning of silicon device layer when removing the
on-top metal, and etching anisotropy.
To understand how the resonance offset alters the frequency conversion process, we simulate the mechanical su-
permodes and calculate the optomechanical and microwave-to-mechanical external coupling rates with a deliberately
introduced geometric offset factor
ξ
to the lattice constant (
ξa
) and the two axes of the ellipse hole (
ξd
1
,
ξd
2
) at
the EMC center (while maintaining the adiabatic tapering curve, the phonon waveguide, and OMC parameters).
The choice of
ξ
= 1 corresponds to the condition of matched EMC and OMC resonances. As shown in fig. 2, for
ξ <
1, the frequency of the EMC breathing mode increases, and aligns spectrally with a parasitic mode localized at
the OMC-phonon waveguide region (where
d
2
and
a
are larger than the OMC center). In this situation, while the
electromechanical coupling remains large, the hybridized mechanical mode is shifted away from the optical cavity
mode. Therefore, the optomechanical coupling rate and consequently the microwave-optical conversion efficiency are
reduced. For
ξ >
1, on the other hand, the frequency of the EMC mode is lower than the original OMC defect mode
and matches with a parasitic mode localized in the OMC-phonon waveguide region (where
d
1
and
a
are larger than
the OMC center, see fig. 3). The poor spatial overlap of the optical and mechanical fields (caused by a spatial shift
in the opposite direction compared to
ξ >
1) results in a reduced optomechanical and microwave-optical conversion
efficiency. We conclude that while the mechanical mode hybridization in our design is robust against disorder, it is
important to fine-tune the EMC/OMC resonances for optimized performance.
The effects of frequency offset can be alternatively understood by keeping track of the breathing mode at the OMC
defect center, which at (
ξ
̸
= 1) weakly hybridize with parasitic mechanical modes in the EMC section, resulting in
supermodes with high optomechanical coupling rates (see the modes near 5.1 GHz in fig. 2 and fig. 3), but weak
electro-mechanical coupling. For example, at
ξ
= 0
.
88, we observe that the OMC defect mode hybridizes with a
shear mode at the EMC section. Since the displacement direction of the shear mode is out of plane, this mode does
not change the air gap of the capacitor and has a negligible
∂C/∂x
(fig. 4a). As another example, at
ξ
= 1
.
06, the
mode at the EMC section is a second-order breathing mode with reduced
∂C/∂x
(fig. 4b). Therefore, while these
modes have significant optomechanical coupling rates, their overall microwave-optical frequency conversion efficiency
4
A
B
B
A
4.9
5
5.1
5.2
5.3
5.4
Frequency (GHz)
0
200
400
600
800
4.9
5
5.1
5.2
5.3
5.4
Frequency (GHz)
0
0.6
1.
2
A
B
Mechanical mode A
Mechanical mode B
Optical mode
a
b
c
g
0
/2
π (kHz)
γ
em
/2
π (Hz)
FIG. 2. Mechanical mode hybridization with the scale parameter set to
ξ
= 0
.
88. (a) Simulated mechanical displacement of
the two primary supermodes and and the electric field of the optical cavity. (b) Calculated optomechanical coupling rates. (c)
Calculated electromechanical decay rates. Dashed lines denote the location of the OMC’s central defect. While the mechanical
hybridization persists in this structure, the optomechanical coupling rates decrease from the optimal design (
ξ
= 1) because of
the spatial misalignment of the mechanical mode with respect to the optical cavity.
4.8
4.9
5
5.1
5.2
5.3
5.4
Frequency (GHz)
0
0.5
1
4.8
4.9
5
5.1
5.2
5.3
5.4
Frequency (GHz)
0
300
600
900
Mechanical mode A
Mechanical mode B
Optical mode
A
B
A
B
a
b
c
g
0
/2
π (kHz)
γ
em
/2
π (Hz)
FIG. 3. Mechanical mode hybridization with the scale parameter set to
ξ
= 1
.
06.(a) Simulated mechanical displacement of
the two primary supermodes and and the electric field of the optical cavity. (b) Calculated optomechanical coupling rates. (c)
Calculated electromechanical decay rates. Dashed lines denote the location of the OMC’s central defect. In addition to the
reduced optomechanical coupling, the EMC/OMC sections are only weakly hybridized, with unequal energy participation in
the two supermodes.
are compromised by the reduced electro-mechanical coupling. Our simulations are in qualitative agreement with the
experimental observations in
??
of the main text.
5
a
b
ξ
= 0.88
ξ
= 1.06
FIG. 4. Mechanical displacement profile of the supermodes with the largest optomechanical coupling for the designs with (a)
ξ
= 0
.
88. (b)
ξ
= 1
.
06.
IV. HOMODYNE DETECTION OF THE MICROWAVE-TO-OPTICAL TRANSDUCTION
We describe the optomechanical interaction in in the rotating frame of the pump laser via the Hamiltonian
ˆ
H/
= ∆ˆ
a
ˆ
a
+
ω
m
ˆ
b
ˆ
b
g
0
ˆ
a
ˆ
a
ˆ
b
(14)
Here, ˆ
a
and
ˆ
b
are the annihilation operators for the optical and mechanical fields. The variables
ω
o
,
ω
m
, and
ω
p
denote the frequencies of the mechanical oscillator, optical cavity and the pump laser, and the detuning parameter is
defined as ∆ =
ω
o
ω
p
. Using this Hamiltonian, the classical response of the system can be derived using a pair of
equations of motions for the classical mode amplitudes
a
=
ˆ
a
and
b
=
ˆ
b
̇
b
=
(
m
+
γ/
2)
b
+
ig
0
a
a
γ
em
b
in
,
(15)
̇
a
=
(
i
∆ +
κ/
2)
a
+
i
(
b
+
b
)
a
κ
e
a
in
(16)
Here,
κ
e
is the optical external coupling from the waveguide coupler to the optical cavity, and
a
in
is the incident
optical field amplitude. Similarly,
γ
em
and
b
in
denote the electromechanical decay rate and the amplitude of the
electrical drive in the microwave waveguide.
For small (optomechanical) cooperativities, the equation for the mechanical mode can be solved by ignoring the
optomechanical interaction, leading to
b
=
n
phon
e
m
t
, with the phonon number given by
??
. In this situation, we
can rewrite the remaining equation for the optical mode as a function of the modulation index
β
= 2
g
0
n
phon
m
̇
a
=
(
i
∆ +
κ/
2)
a
+
iβω
m
cos (
ω
m
t
)
a
κ
e
a
in
(17)
For a small
β
in the sideband resolved regime (
κ < ω
m
), only the first-order sidebands are pertinent in the intracavity
optical field. Hence, it is appropriate to write the optical field in the rotating frame of the laser carrier frequency as
a
=
a
1
e
m
t
+
a
0
+
a
1
e
m
t
(18)
6
Plugging eq. (18) into eq. (17), we have, for each frequency component,
a
0
=
κ
e
i
∆ +
κ/
2
a
in
(19)
a
1
=
iβω
m
/
2
i
(∆
ω
m
) +
κ/
2
a
0
(20)
a
1
=
iβω
m
/
2
i
(∆ +
ω
m
) +
κ/
2
a
0
(21)
The optical waveguide output field can be written as a function of the field inside the cavity as
a
out
=
a
in
+
κ
e
a
=
A
0
A
1
e
m
t
A
1
e
+
m
t
.
(22)
For the cases when the laser pumps is detuned by one mechanical frequency to the red or blue side of the optical
cavity (∆ =
±
ω
m
), the modulation creates predominantly a single frequency component
A
±
1
=
iβω
m
κ
κ
e
±
m
+
κ/
2
a
in
.
(23)
The microwave-to-optical power conversion efficiency can be written as the ratio of the power in the generated optical
side-bands normalized to power of the electrical drive used to excite the mechanical mode
P
o
(
ω
m
)
P
rf
=
|
A
±
1
|
2
P
rf
=
β
2
ω
2
m
κ
2
κ
2
e
ω
2
m
+
κ
2
/
4
P
in,o
,
(24)
where
P
in,o
=
|
a
in
|
2
is the optical pump power at the feed waveguide. Using the definition of
V
π
(the peak microwave
voltage required to excite the mechanical mode sufficiently for achieving a modulation index of
β
=
π
), the modulation
index can be substituted as
β
=
π
p
2
Z
0
P
rf
/V
2
π
in the expression for the efficiency
P
o
(
ω
m
)
P
rf
=
2
π
2
Z
0
ω
2
m
κ
2
e
κ
2
(
ω
2
m
+
κ
2
/
4)
V
2
π
P
in,o
,
(25)
where
Z
0
is the impedance of the transmission line. Subsequently, we recast the power conversion efficiency the photon
flux conversion efficiency
P
o
(
ω
m
)
/
ω
p
P
rf
/
ω
m
=
2
π
2
Z
0
ω
3
m
κ
2
e
κ
2
ω
p
(
ω
2
m
+
κ
2
/
4)
V
2
π
P
in,o
(26)
At low intra-cavity photon numbers (
n
c
<< κγ/
4
g
2
0
), eq. (26) is equivalent to
η
oe
= 4
C
em
C
om
/
(1+
C
em
+
C
om
)
2
(barring
the extraction factor
η
o
=
κ
e
) [4]. Note that this conversion efficiency is inversely proportional to
V
2
π
.
We use the homodyne setup fig. 5 to detect the converted optical frequency component. We ensure a low modulation
index by setting the DC-bias voltage at
V
b
= 10 V and microwave drive power at
P
rf
=
6 dBm. When the laser
incident is at the blue sideband, the converted frequency component beats with the laser frequency at the high-speed
photodetector, resulting in the microwave voltage received by the vector network analyzer (VNA)
|
s
o
(
ω
m
)
|
=
2
βω
m
κ
e
κ
p
ω
2
m
+
κ
2
/
4
G
·
P
in,o
(27)
where the factor
G
includes the power amplification of the EDFA, power-to-voltage response of the photodetector,
optical fiber loss and microwave cable loss. The magnitude of the
S
21
trace measures the voltage in eq. (25) over the
incident microwave voltage
|
S
21
|
=
|
s
o
(
ω
m
)
|
V
in
=
2
2
ω
m
κ
e
V
π
κ
p
ω
2
m
+
κ
2
/
4
G
·
P
in,o
(28)
Comparing eq. (28) with eq. (25), we conclude that
|
S
21
|
is proportional to the square-root of the frequency conversion
efficiency at any given laser power
in,o
. Therefore, we use the experimental measurements of
|
S
21
|
to characterize the
spectra of transduction in our devices.
7
VNA
Bias T
+
-
V
b
P1
P2
Laser
Circulator
HPD
EDFA
RSA
VOA
Polarization
controller
LPD
BPF
OS
FIG. 5. Detailed schematics of the measurement setup for the characterizing electro-optomechanical frequency conversion.
VOA: variable optical attenuator; LPD: low-speed photodetector. This detector is used for measuring the optical reflection
spectrum; OS: optical switch; HPD: high-speed photodetector. This detector has a 25 GHz bandwidth and is used for measuring
the modulated sidebands.; EDFA: erbium-doped fiber amplifier; BPF: optical band-pass filter; VNA: vector network analyzer;
RSA: real-time spectrum analyzer.
V. FITTING OF THE REFLECTION SPECTRA AT HIGH MODULATION INDICES
For a high modulation index (
β >
1), higher-order sidebands are involved in the solution of eq. (17), which leads
to splitting of the reflection spectrum. Here, we derive the general solution of eq. (17) and show how
β
can be obtain
from the reflection spectrum fitting.
By the transformation
a
(
t
) =
α
(
t
) exp
{
[
sin (
ω
m
t
+
φ
)]
}
, we rewrite eq. (17) as
̇
α
=
(
i
∆ +
κ/
2)
α
κ
e
e
sin (
ω
m
t
+
φ
)
a
in
(29)
Using Jacobi–Anger expansion
e
sin (
ω
m
t
+
φ
)
=
X
k
J
k
(
β
)
e
ik
(
ω
m
t
+
φ
)
(30)
where
J
k
(
β
) is the Bessel function of the first kind, and
α
(
t
) =
P
k
α
k
exp
{
(
ikω
m
t
)
}
we transform eq. (29) to find Eq
S22urier coefficients
ikω
m
α
k
=
(
i
∆ +
κ/
2)
α
k
J
k
(
β
)
e
ikφ
κ
e
a
in
(31)
which leads to
α
k
=
e
ikφ
J
k
(
β
)
κ
e
i
(∆ +
m
) +
κ/
2
a
in
(32)
The intracavity optical field is thereby the convolution
a
(
t
) =
X
n
e
in
(
ω
m
+
φ
)
X
k
J
n
+
k
(
β
)
J
n
(
β
)
κ
e
a
in
i
(∆ +
m
) +
κ/
2
(33)
from which we can calculate the reflected optical field
a
out
(
t
) =
a
in
+
κ
e
a
(
t
)
(34)
8
We measure the spectra of the reflected optical power via a low-speed photodetector with the maximum bandwidth
of 10 MHz which only detects the slowly-varying envelope of the optical field. Therefore, the reflection spectrum at
the low-speed photodetector is
R
=

|
a
out
|
2
|
a
in
|
2

=
X
k
J
k
(
β
)
J
k
(
β
)
κ
e
i
(∆ +
m
) +
κ/
2
2
(35)
where
⟨·⟩
denotes time averaging (due to the small bandwidth of the detector). In our fabricated device, however,
the total optical reflection includes stray light reflection from non-resonance structures, which contributes to a static
noisy background. Before fitting the measured reflected power spectra to eq. (34), it is necessary to remove the
background features. Since the background is invariant under different modulations of the OMC optical cavity, we
extract the static background by interpolating the optical resonance of the reflectance spectrum without modulation.
The backgrounds of the modulated spectra are thereby removed by normalizing the spectra by the obtained non-
modulated background. After the background removal, we are able to fit the experimental data using eq. (34) for
each plot in Fig. 4e in the main text and extract the corresponding modulation index.
VI. LIMIT OF THE DC-BIAS VOLTAGE
The maximum electromechanical decay rate is set by the maximum DC voltage that can be applied before the
breakdown of our device. This upper limit is set by the pull-in voltage at which the nanobeam (the center electrode)
touches one of the outer ground electrodes. This ‘pull-in’ phenomenon is commonly observed in electrostatic actuators
when the electrostatic force with increasing voltage becomes too strong to be reset by the effective mechanical spring
force, leading to unstable mechanical dynamics. Once the bias voltage reaches the onset of such instability, the
electrodes will not recover the original positions due to the static stiction (see fig. 6a). The pull-in is occasionally
accompanied by permanent structural damage, which is suspected to be caused by a transient large current through
the shut-down capacitor, leading to heat generation and the meltdown and collapsing of the mechanical structure (see
fig. 6b). We have measured a repeatable breakdown voltages of 15
±
1V across 5 devices.
2.5 μm
2.5 μm
a
b
FIG. 6. SEM images of (a) a nanobeam touching one of the ground electrodes featuring the pull-in phenomenon and (b) a
broken-down device with collapsed nanobeam after applying a DC-voltage of 14 V. The voltage that results in the pull-in of the
device in (a) is 16V. The device does not melt, which may be attributed to an out-of-plane motion without a direct touching
of the metal layer. The reshaping of the phonon shields near the electrodes and the peeling-off of the TiN layer suggest the
damage was created by heating from the short circuit.
9
VII. COMPARISON WITH PREVIOUS WORK
We have compared our device’s performance with previous room-temperature implementations in different piezo-
electric material platforms in table I. We note that the reported conversion efficiencies in the literature are obtained
at different optical pump powers and external fiber-to-chip coupling efficiencies, parameters that are not intrinsically
related to the material properties and the devices’ performance. To make a direct comparison possible, we factor out
these effects by calculating the intrinsic microwave-to-optical conversion efficiency from the reported optomechanical
and electromechanical cooperativities using the expression
η
oe
= 4
C
em
C
om
/
(1 +
C
em
+
C
om
)
2
at the intracavity photon
number
n
c
= 1. We estimate the electromechanical cooperativity in Balram et al. [5] by using their reported op-
tomechanical coupling rates and the half-wave voltage. Conversely, we estimate the half-wave voltages in Vainsencher
et al. [6] and Marinkovi ́c et al. [7] from their reported mechanical frequencies, optomechanical coupling rates, and
electromechanical decay rates.
References
Vainsencher,
et al [6]
Balram,
et
al [5]
Jiang, et al [8]
Marinkovi ́c,
et al [7]
This work
g
0
/
2
π
(kHz)
110
1100
80
10
577
κ/
2
π
(GHz)
15
5
.
2
1
.
2
6
.
6
1.4
γ/
2
π
(MHz)
5
1
.
7
1
.
9
12
3.5
C
om
(
n
c
= 1)
6
.
5
×
10
7
5
.
4
×
10
4
1
.
2
×
10
5
5
×
10
9
2
.
7
×
10
4
C
em
6
×
10
4
1
.
2
×
10
8
10
3
0
.
015
4
.
8
×
10
7
η
oe
(
n
c
= 1)
1
.
7
×
10
9
2
.
6
×
10
11
4
.
5
×
10
8
2
.
94
×
10
10
5
.
2
×
10
10
V
π
(V)
0.1
0
.
65
0
.
024
0.16
0.75
Platform
AlN
GaAs
LN
LN-SOI
SOI
TABLE I. Comparison of room-temperature experiments in our electro-optomechanical transducer and piezo-optomechanical
devices across different material platforms.
η
oe
represents the on-chip single-pump-photon efficiency. The starred quantities
are reported in the references, and the rest are calculated (see the main text).
g
0
/
2
π
(kHz)
κ/
2
π
(GHz)
κ
mic
/
2
π
(MHz)
γ/
2
π
(kHz)
Z
r
(kΩ)
n
c
C
om
C
em
(
V
b
= 10 v)
η
oe
Bandwidth
(kHz)
577
1.4
1
30
3.5
110
3.4
16.6
0
.
52
630
TABLE II. Estimated transducer parameters at 20 mK. .
The data in this table shows that our silicon-based device operates comparably to most piezo-optomechanical
devices and falls significantly short only compared to previous work with LiNbO
3
[8]. The primary reason for the
lower efficiency of our device compared to this platform is the smaller electromechanical cooperativity, which is limited
by the high impedance of our transducer and the pull-in voltage. Despite these limitations, our device’s performance
is expected to remain competitive at cryogenic temperatures, as discussed in VIII.
VIII. PERFORMANCE AT CRYOGENIC TEMPERATURES
Operation at lower temperatures is expected to affect our device parameters in two major ways. Firstly, utilizing
a high-impedance microwave resonator between the mechanical resonator and the waveguide is expected to result in
a significant increase in the electromechanical decay. Using conservative estimates for the impedance values (
Z
r
=
3
.
5 kΩ) and moderate quality factors of
Q
= 5000 (for previous demonstrations exceeding these values, see [9]), we
anticipate an enhancement factor of up to 5
×
10
5
in the electromechanical decay rate (see eq. (13)). Additionally,
the mechanical mode is expected to become significantly narrower, decreasing from its current value of 3-4 MHz to
sub-100 kHz levels, based on previous measurements in silicon optomechanical devices [10, 11]. We use a conservative
value of 30 kHz for the mechanical mode width for our calculations.
Table II presents the list of assumed and calculated parameters. Therein, the bandwidth is determined by the
optomechanical and electromechanical decay rates, as well as the mechanical linewidth in the following way
B
= (
γ
em
+
γ
om
+
γ
)
/
2
π
(36)
At millikelvin temperatures, the linewidth of the mechanical modes decreases significantly, and the contributions from
the optomechanical and electromechanical decay rates become dominant. As shown, we estimate that an internal
conversion efficiency beyond 50% and a bandwidth exceeding 500 kHz is attainable with reasonable parameters. The