Efficient microwave frequency conversion mediated
by the vibrational motion of a silicon nitride
nanobeam oscillator
J. M. Fink
1
,
2
,
3
, M. Kalaee
1
,
2
R. Norte
1
,
2
‡
, A. Pitanti
1
,
2
§
and
O. Painter
1
,
2
1
Kavli Nanoscience Institute and Thomas J. Watson, Sr., Laboratory of
Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA
3
Institute of Science and Technology Austria (IST Austria), 3400
Klosterneuburg, Austria
E-mail:
jfink@ist.ac.at
Septempber 2018
Abstract.
Microelectromechanical systems and integrated photonics provide
the basis for many reliable and compact circuit elements in modern
communication systems. Electro-opto-mechanical devices are currently one of
the leading approaches to realize ultra-sensitive, low-loss transducers for an
emerging quantum information technology. Here we present an on-chip microwave
frequency converter based on a planar aluminum on silicon nitride platform that
is compatible with slot-mode coupled photonic crystal cavities. We show efficient
frequency conversion between two propagating microwave modes mediated by the
radiation pressure interaction with a metalized dielectric nanobeam oscillator. We
achieve bidirectional coherent conversion with a total device efficiency of up to
∼
60%, a dynamic range of 2
×
10
9
photons/s and an instantaneous bandwidth
of up to 1.7 kHz. A high fidelity quantum state transfer would be possible if the
drive dependent output noise of currently
∼
14 photons
·
s
−
1
·
Hz
−
1
is further
reduced. Such a silicon nitride based transducer is in-situ reconfigurable and could
be used for on-chip classical and quantum signal routing and filtering, both for
microwave and hybrid microwave-optical applications.
Keywords
:
superconducting circuits, electromechanics, optomechanics, MEMS,
frequency conversion, hybrid devices, silicon nitride membranes
‡
Present address: Department of Precision and Microsystems Engineering, Delft University of
Technology, Mekelweg 2, 2628CD Delft, The Netherlands
§
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, piazza San Silvestro 12, 56127
Pisa (PI), Italy
arXiv:1911.12450v1 [quant-ph] 27 Nov 2019
Microwave frequency conversion with a silicon nitride nanobeam oscillator
2
1. Introduction
Silicon nitride (Si
3
N
4
) thin films show exceptional
optical and mechanical properties [1], and are used
in many microelectromechanical and photonic devices.
The material’s large bandgap [2], high power handling
due to the absence of two-photon absorption in the
telecom band [3] and the low absorption losses in
Si
3
N
4
thin films [4] make it an ideal candidate for
many photonics applications, ranging from nonlinear
optics [5, 6], to atom trapping [7, 8] and tests of
quantum gravity [9]. The structural stability and high
mechanical quality factor [10] of high tensile stress
Si
3
N
4
thin films grown by low-pressure chemical vapor
deposition enables nanostructures to be patterned
with extreme aspect ratios [11] and form up to
centimeter scale patterned membranes [12, 13] with
high reflectivity [14, 15]. New soft clamping techniques
make use of the tensile stress to maximize the
mechanical quality factor [16, 17, 18], allowing for
an unprecedented regime in which quantum coherence
can be reached for micromechanical systems even in
a room temperature environment [19]. Additionally,
slot mode 1D photonic crystal cavities have been
developed using Si
3
N
4
thin films to realize strong
optomechanical interactions in a small mode volume
and fully integrated on-chip [20, 21, 22, 23].
In the microwave domain, Si
3
N
4
is widely used for
wiring capacitors and cross-overs. Early work focussed
on the study of Si
3
N
4
as a low loss dielectric to realize
compact capacitive circuit elements operated in the
quantum regime [24]; however, the amorphous material
and its surface are known to host two-level defects
[25], such as hydrogen impurities with sizable dipole
moments and life-times, which led to the observation of
strong coupling between a single two-level system and
a superconducting resonator [26, 27, 28]. Nonetheless,
due to its unique mechanical properties, high quality
factor membranes [29, 30] as well as micro-machined
Si
3
N
4
nanobeams [31] have been coupled capacitively
to superconducting resonators [32, 33] in the context
of cavity electromechanics. In the latter experiments
the achievable coupling strength was fundamentally
limited by the small participation ratio of the motional
capacitance. We recently presented a new platform
that uses the membrane itself as a low loss substrate
for the microwave resonator, drastically lowering
the parasitic circuit capacitance and maximizing
the electromechanical coupling between a metalized
silicon nitride nanobeam and a high impedance
superconducting coil resonator.
This allowed for
high cooperativities and successfully demonstrating
sideband cooling of the low MHz frequency nanobeam
to the motional ground state [27].
Silicon nitride membrane-based devices are cur-
rently the leading approach to couple optical and mi-
crowave systems [34, 35]. Realizing noiseless conversion
with a mechanical oscillator [36, 37, 38, 39, 40, 41, 42]
would allow one to build transducers for quantum net-
works of superconducting processors connected via re-
silient and low loss optical fiber networks. Efficient
wavelength conversion has been realized between op-
tical wavelengths using silicon optomechanical crys-
tals [43], between microwave frequencies using metal-
lic drum resonators [44, 45, 46] and silicon nanobeams
[47], and also between microwave and optical wave-
length using silicon nitride membrane based Fabry-
Perot cavities [34, 35].
Alternative approaches in-
clude the use of Josephson circuits for conversion in
the microwave domain [48, 49, 50], Bragg scattering
in silicon nitride rings [6] and dispersion engineering
of silicon nitride waveguides [2] in the optical domain.
Coupling RF and microwave fields to optics has been
achieved with membranes [51, 52], via a mechanical
intermediary in combination with the piezoelectric ef-
fect and optomechanical interactions [53, 54, 55, 56],
and microwave to optics conversion has been proposed
[57, 58, 59] and realized with high bandwidth via the
electro-optic effect [60, 61].
It is an outstanding challenge to realize an on-
chip integrated microwave to optics converter based
on the radiation pressure (optomechanical) interaction
alone (i.e., on both the microwave and optical
side of the converter).
Mechanical systems offer
the potential to fully separate the sensitive optical
modes (superconductors cause optical loss) from the
equally sensitive superconducting circuits (optical
light generates quasi-particles in superconductors); for
example using phononic waveguides. Here we present
a coherent microwave frequency converter on the
aluminum-on-Si
3
N
4
platform [27] that is compatible
with on-chip optomechanics [21, 22]. In the future this
approach could be used to realize conversion between
microwave and optical fields, or to implement low
voltage modulation and fully electrical tunability in
Si
3
N
4
-based photonic devices.
2. Implementation
2.1. Physics
We realize a system where one mechanical oscillator
mode with frequency
ω
m
and damping rate
γ
m
is
coupled to two electromagnetic resonator modes with
resonance frequencies
ω
i
and linewidths
κ
i
(
i
=
{
1
,
2
}
)
via the optomechanical radiation pressure interaction
as proposed in Refs. [37, 38, 39]. In the presence of
two red detuned classical drive fields
α
d
,i
near the red
sideband of the respective microwave mode at
ω
d
,i
=
ω
i
−
ω
m
the parametric interaction can be linearized
and described by the sum of two beam splitter type
interactions that allow to swap excitations between
Microwave frequency conversion with a silicon nitride nanobeam oscillator
3
the mechanical and the two electromagnetic modes, see
Fig. 1(a). In the resolved-sideband limit (
ω
m
κ
i
,γ
m
)
the linearized electromechanical Hamiltonian in the
rotating frames and the rotating wave approximation
is given by
H
=
∑
i
=1
,
2
̄
h
∆
i
ˆ
a
†
i
ˆ
a
i
+ ̄
hω
m
ˆ
b
†
ˆ
b
+
∑
i
=1
,
2
̄
hg
i
(
ˆ
a
i
ˆ
b
†
+
ˆ
b
ˆ
a
†
i
)
,
(1)
where ˆ
a
i
is the annihilation operator for the microwave
field mode,
ˆ
b
is the annihilation operator of the
mechanical mode, ∆
i
=
ω
i
−
ω
d
,i
=
ω
m
is the
detuning between the external driving field and the
relevant resonator resonance, and
g
i
=
g
0
,i
√
n
i
is the electromechanical coupling strength between
the mechanical mode and resonator
i
with
n
i
=
|
α
d
,i
|
2
=
P
in
,i
̄
hω
d
,i
4
κ
ex
,i
κ
2
i
+4∆
2
i
the number of intra-resonator
drive photons for the microwave input power with
P
in
,i
.
The interaction terms of the Hamiltonian in
Eq. (1) have two closely related effects. Optomechani-
cal damping cools the mechanical motion with the rate
Γ
i
=
4
g
2
i
κ
i
. At the same time this leads to the desired
bidirectional photon conversion between two distinct
electromagnetic frequencies. Using input-output the-
ory, we can relate the itinerant input and output modes
to the intra-cavity modes as ˆ
a
out
,i
=
√
κ
ex
,i
ˆ
a
i
−
ˆ
a
in
,i
.
In the photon conversion process, an input microwave
signal at frequency
ω
s
,
1
with amplitude ˆ
a
in
,
1
is down-
converted to the mechanical mode at frequency
ω
m
,
which corresponds to
ˆ
b
†
ˆ
a
1
in Eq. (1). Next, during
an up-conversion process the mechanical mode trans-
fers its energy to the output of the other microwave
resonator at frequency
ω
2
and amplitude ˆ
a
out
,
2
, which
corresponds to ˆ
a
†
2
ˆ
b
in Eq. (1). In this process the me-
chanical resonance is virtually populated, in the sense
that the input signal is rapidly converted to the out-
put signal, leaving little time for the population of the
intermediate mechanics. Likewise, an input microwave
signal at frequency
ω
2
can be converted to frequency
ω
1
by reversing the conversion process, see Fig. 1(a). The
Hermitian aspect of the Hamiltonian (1) makes this
process bidirectional, without any unwanted loss, gain
or noise.
We define the photon conversion efficiency via
the transmission scattering parameter, i.e. as the
ratio of the output-signal photon flux over the input-
signal photon flux,
|
S
ij
|
2
=
∣
∣
∣
ˆ
a
out
,i
ˆ
a
in
,j
∣
∣
∣
2
. By solving the
linearized Langevin equations we find that for signals
on resonance with the microwave resonator, in the
steady state the bidirectional conversion efficiency is
given as [62]
|
S
ij
|
2
=
|
S
ji
|
2
=
|
T
|
2
=
η
i
η
j
4
C
i
C
j
(1 +
C
i
+
C
j
)
2
,
(2)
with
i,j
=
{
1
,
2
}
the indices of the two modes.
C
i
=
Γ
i
γ
m
is the electromechanical cooperativity for resonator
C
m,1
C
s,1
L
1
a
in,1
a
out,2
L
2
C
s,2
C
m,2
(b)
(a)
!
s,1
!
d,1
!
1
≈
2
¼
7.4 GHz
!
m
!
s,2
!
d,2
!
2
≈
2
¼
9.3 GHz
!
m
S
21
,
S
12
100
¹
m
20
¹
m
2
¹
m
2
¹
m
C
m,1
L
1
L
2
C
m,2
(d)
!
d,1
!
d,2
!
s
VNA
SA
a
in,1
a
out,2
S
21
a
in,2
a
out,1
S
12
a
in,1
a
out,1
S
11
a
in,2
a
out,2
S
22
®
1,
®
2
̄
1,
̄
2
(c)
^
^
^
^
^
^
^
^
^
^
Figure 1.
(a), Schematic presentation of the frequency
conversion.
The spectral density of the two microwave
resonators at frequencies
ω
i
(black lines), the strong drive
tones at frequencies
ω
d
,i
=
ω
i
−
ω
m
(long red and blue
arrows) and the signal tones at the optimal frequencies
ω
s
,i
=
ω
i
(short red and blue arrows) for
i
=
{
1
,
2
}
,
as well as the conversion scattering parameters
S
21
and
S
12
are indicated.
(b), Circuit diagram of the converter.
The silicon nitride nanobeam in-plane fundamental mode
displacement (color indicates displacement amplitude) is coupled
capacitively via its two modulated capacitances
C
m
,i
to two
parallel inductance-capacitance resonators realized with high
characteristic impedance planar spiral inductors with the
inductances
L
i
and the stray capacitances
C
s
,i
.
The two
resonant circuits are coupled inductively to a transmission line
to couple in and out the propagating microwave modes ˆ
a
in
,i
and ˆ
a
out
,i
. (c), False color scanning electron micrograph of the
converter device with thin-film aluminum (white) on suspended
silicon nitride membrane (blue). Mechanical beam, cross-over
and capacitor region are shown enlarged. (d), Experimental
setup. Three microwave sources and one vector network analyzer
(VNA) output are combined at room temperature, attenuated
by
α
i
and coupled to the device at about 12 mK using semirigid
coaxial cables, a low loss printed circuit board and an on-
chip coplanar waveguide.
The reflected signals at the two
frequencies of interest are routed to the output path using
a cryogenic circulator and after passing another isolator (not
shown) are amplified by
β
i
at the 4 Kelvin stage and also at room
temperature before detection with either a spectrum analyzer
(SA) or the VNA input.
i
and
η
i
=
κ
ex
,i
κ
i
is the waveguide-to-resonator coupling
ratio with
κ
i
=
κ
in
,i
+
κ
ex
,i
the total damping rate,
κ
ex
,i
the decay rate into the waveguide and
κ
in
,i
the
decay rate to any other mode. We also obtain a simple
equation for the two reflection coefficients, which are
Microwave frequency conversion with a silicon nitride nanobeam oscillator
4
given by
|
S
ii
|
2
=
(
1
−
2
η
i
(1 +
C
j
)
1 +
C
i
+
C
j
)
2
.
(3)
For lossless microwave cavities
η
i
= 1 and in the
limit
C
1
=
C
2
1 near unity photon conversion
efficiency with
|
T
|
2
= 1 and
|
S
11
|
2
=
|
S
22
|
2
=
0 (zero reflection) can be achieved.
The former
condition (
C
1
=
C
2
) balances the photon-phonon
conversion rates Γ
i
, while the latter condition (
C
i
1) guarantees the mechanical damping rate is much
smaller than the conversion rates
γ
m
Γ
i
. In this
limit the photon-to-photon conversion rate exceeds
the mechanical damping rate - the rate at which
phonons are exchanged with the noisy environment.
This conversion process is coherent with the bandwidth
given by Γ =
γ
m
+ Γ
1
+ Γ
2
, which is the total back-
action-damped linewidth of the mechanical resonator
in the presence of the two microwave drive fields.
2.2. Circuit
We implement bidirectional frequency conversion in
a circuit as shown in Fig. 1(b). The two microwave
resonators with resonance frequencies
ω
1
= 7
.
444 GHz
and
ω
2
= 9
.
308 GHz are realized using two lumped
element inductor-capacitor (LC) circuits formed from
a planar spiral inductor of high impedance.
The
capacitance of these lumped element resonant circuits
is defined by the sum of the stray capacitance of
the circuit, which is dominated by the inductor
stray capacitance, and the mechanically modulated
capacitance.
The two resonators are inductively
coupled to a single physical port - a 50 Ω coplanar
waveguide that is shorted to ground using a thin
superconducting wire close the the two inductors.
2.3. Device
The described circuit is fabricated on the aluminum-
on-Si
3
N
4
platform similar to Ref. [27]. Here the entire
aluminum circuit, which is shown in Fig. 1(c), is
suspended on a fully under-etched high-stress Si
3
N
4
membrane on a high resistivity silicon chip.
The
inductors are realized as planar spiral inductors with
a pitch of 1
μ
m which maximizes the obtained
geometric inductance per unit length, and together
with the small effective permittivity of the 60 nm
thin membrane, minimizes the stray capacitance of
the circuit.
This in turn maximizes the obtained
electromechnical couplings yielding measured values
of
g
0
,
1
/
2
π
= 33 Hz and
g
0
,
2
/
2
π
= 44 Hz for
the fundamental in-plane mechanical mode of the
patterned silicon nitride nanobeam with an intrinsic
damping rate of
γ
m
/
2
π
= 7 Hz at a resonance
frequency of
ω
m
/
2
π
= 4
.
118 MHz. This is in good
agreement with calculations based on perturbation
theory and electromagnetic modeling of the electric
field strength at the dielectric and metallic boundaries
of the vacuum gap capacitor [63]. While not measured
in this work, the Si
3
N
4
nanobeam has been designed as
a phononic bandgap crystal that also localizes a high
frequency acoustic defect mode [27]. Very recently,
quantum-level transduction of hypersonic mechanical
motion could be demonstrated with a similar device
[64].
2.4. Setup
The experiment is performed at 12 mK inside a dilution
refrigerator. At room temperature we apply the drive
and signal tones with low noise microwave sources and
detect both reflection scattering parameters with a vec-
tor network analyzer (VNA) and the two transmission
scattering parameters with a spectrum analyzer (SA),
as shown in Fig. 1(d). Inside the dilution refrigerator
we distribute 50 Ω attenuators at various temperature
stages to thermalize the electromagnetic mode temper-
ature with the refrigerator temperature. We use one
circulator to couple to the single physical port of the
device in a reflection geometry. A second isolator is
used to isolate the device from noise at 4 Kelvin where
a commercial low noise HEMT amplifier is positioned.
The four scattering parameters
S
ii
between the two
mode frequencies
ω
i
presented in this work refer to the
ratios of the 4 propagating modes in a single on-chip
waveguide as schematically shown in Fig. 1(d).
3. Characterization
3.1. Resonators
As a first step we characterize the resonator properties
using a VNA. The magnitudes of the measured
complex reflection coefficients
S
ii
are normalized with
α
i
β
i
→
1, which now corresponds to the scattering
parameter at the position of the on-chip waveguide.
Then the measured in-phase and quadrature phase
components are fitted to the real and imaginary
components of
S
ii
(
ω
) =
e
−
i
(
φ
+
ωτ
)
(
1
−
κ
ex
,i
κ
i
/
2 +
i
(
ω
i
−
ω
)
)
,
(4)
where
φ
is a global phase offset and
τ
≈
50 ns is
the delay of the signal in our setup. The result for
both resonator modes are shown in Fig. 2(a). Here
we plot the magnitude and phase of the measurement
(blue points) and the fit (red lines) with excellent
agreement. One can see that the resonators are both
over-coupled, i.e.
κ
ex
,i
> κ
in
,i
as indicated by the
full phase shift of
∼
2
π
. The comparably very high
quality factors of
Q
in
,i
=
ω
i
κ
in
,i
=
{
2
.
2
×
10
5
,
5
.
5
×
10
4
}
Microwave frequency conversion with a silicon nitride nanobeam oscillator
5
enable the large waveguide coupling constants of up
to
η
i
=
{
0
.
92
,
0
.
68
}
that are essential for the efficient
conversion process (c.f. Eq. (2)). In general, we find
the intrinsic losses to be drive power dependent, likely
due to saturation of two-level system absorption in the
amorphous Si
3
N
4
[24]. For the powers studied in this
manuscript we determined coupling efficiencies in the
range of
η
i
=
{
0
.
80
−
0
.
92
,
0
.
54
−
0
.
68
}
.
3.2. Two-mode EIT
As a second step we study the reflection scattering
parameters measured with a weak probe tone using
the VNA in the presence of two strong red-
detuned drive tones. Here we observe a variant of
optomechanically induced transparency [65, 66, 67],
an analog to electromagnetically induced transparency
(EIT), where the mechanical sideband generated from
the drives by the optomechanical interaction interfere
with the weak probe tone from the VNA to modify
the coherent resonator spectrum.
In the case of
two resonators and drive tones interacting with one
mechanical mode we can model [68] and fit the
measurements with
S
ii
(
ω
) = 1
−
κ
ex
,i
χ
r
,i
(
1 +
g
2
j
χ
m
χ
r
,j
)
1 +
χ
m
∑
i
g
2
i
χ
r
,i
(5)
with
i,j
=
{
1
,
2
}
the indices of the resonator modes,
χ
−
1
r
,i
=
κ
i
/
2 +
i
(∆
i
−
ω
) the resonator susceptibilities
and
χ
−
1
m
=
γ
m
/
2 +
i
(
ω
m
−
ω
) the mechanical
susceptibility. Figure 2(b) shows a measurement (blue
points) of the reflection scattering parameter in the
vicinity of the two resonator modes together with a fit
to Eq. (5) (red lines).
Similar to the case of standard EIT-type measure-
ments, the formation of the peak or dip due to the me-
chanical mode with total linewidth Γ (as indicated in
the insets of Fig. 2(b)) depends on the level of coop-
erativity
C
and the degree of coupling
η
. However, in
the present case the interpretation is more complicated
because there are two terms that cause optomechani-
cal damping. They affect each other and the double-
EIT spectrum as a whole. For the chosen pump power
and cooperativity combination, which is indicated in
Fig. 2(c) and (d) by red data points, we observe a full
suppression of the reflected probe signal for resonator
mode 1 Fig. 2(b). This is a necessary condition for high
efficiency conversion. In the case of resonator mode 2
on the other hand, which is only critically coupled to
the waveguide (
η
∼
0
.
5), we observe a peak in the cen-
ter of the resonator, indicating a finite reflection that
results in a limited conversion efficiency.
3.3. Cooperativity
We performed two-mode EIT measurements and
analyses as presented above for all of the following
(a)
(c)
(d)
(e)
(f)
7.4425
7.4435
7.4445
7.4455
0.0
1.0
!
/2
¼
(
GHz
)
!
/2
¼
(
GHz
)
|
S
ii
|
2
9.306
9.307
9.308
9.309
-
¼
¼
Arg
(
S
ii
)
Q
ex,1
Q
ex,2
¼
2.0
10
4
Q
in,1
Q
in,2
¼
2.2
10
5
¼
2.6
10
4
¼
5.5
10
4
·
·
·
·
(b)
7.443
7.444
7.445
0
1
9.307
9.308
9.309
j
S
ii
j
2
!
/2
¼
(
GHz
)
¡
¡
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
□
□
□
□
□
□
□
□
□
□
□
□
-
20
-
10
0
10
0.5
1
5
10
10
3
10
4
10
5
10
6
0.2
0.5
1
2
5
n
r
,1
,
n
m
,1
n
d
,1
T
m
,1
(
mK
)
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
◦
□
□
□
□
□
□
□
□
□
□
□
□
-
20
-
10
0
10
1
5
10
50
10
3
10
4
10
5
10
6
0.2
0.5
1
2
5
n
r
,2
,
n
m
,2
n
d
,2
T
m
,2
(
mK
)
-
12
-
8
-
4
0
5
10
50
100
5
·
10
3
10
4
5
·
10
4
10
5
50
100
500
1000
P
(
dBm
)
C
1
n
d,1
d,1
d,2
¡
,1
¡
,2
/
2
π (
Hz
)
-
8
-
4
0
5
10
50
100
5
·
10
3
10
4
5
·
10
4
50
100
500
1000
(
dBm
)
C
2
n
d,2
/
2
π (
Hz
)
P
P
(
dBm
)
d,1
d,2
(
dBm
)
P
Figure 2.
(a), Normalized reflected power (left axis) and
phase (right axis) of a VNA measurement of the two microwave
resonator modes (blue points) and a combined fit to the real
and imaginary part of Eq. (4). The fitted extrinsic and intrinsic
resonator quality factor are indicated.
(b), Two-mode EIT
measurement in the presence of two drive tones at
ω
d
,
1
and
ω
d
,
2
(blue points) and combined fit to the squared norm of
Eq. (5) for the two microwave modes. Insets show an enlarged
view in the frequency direction (boxed area of main plot) of the
mechanical response with identical total linewidth Γ. (c), and
(d), Fitted optomechanical damping rate Γ
i
and cooperativity
C
i
as a function of the drive power on resonator 1 (panel c)
and 2 (panel d). Error bars show the standard deviation of
the fitted cooperativity of one resonator inferred from multiple
measurements with different drive powers applied to the other
resonator. The data and fit shown in panel (b) was used for the
data points indicated with red color. Panels (e), and (f), show
the mechanical
n
m
,i
(blue squares) and resonator occupations
n
r
,i
(red circles) as well as the mechanical noise temperatures
T
m
,i
(right axis) as a function of red detuned drive power and
the drive photon number.
pump power combinations:
P
d
,
1
from -14 dBm to
0 dBm and
P
d
,
2
from -10 dBm to 2 dBm, both in
steps of 2 dB. Here the input power at the device
P
in
,i
=
P
d
,i
−
α
i
(on a log scale) is related to the shown
drive powers via the attenuations
α
i
=
{
69
.
0
,
70
.
4
}
dB
for the two mode frequencies of interest. For each
power combination we fit both spectra to a single set of
parameters (specifically
g
i
) and summarize the results
Microwave frequency conversion with a silicon nitride nanobeam oscillator
6
in Fig. 2(c) and (d). Shown are the mean and the
statistical error of the cooperativities
C
i
=
Γ
i
γ
m
. Using
γ
m
/
2
π
= 7 Hz obtained from cooling measurements
discussed below, we also back out the optomechanical
damping Γ
i
=
4
g
2
i
κ
i
for each of the drive powers
P
d
,i
,
and the intra-resonator drive photon numbers
n
d
,i
.
We find that the dependence on drive power follows
the expected behavior (dashed lines) based on the
applied drive powers, attenuations and the calibrated
g
0
,i
.
The small error bars confirm that
C
1
is not
significantly affected by changing
P
d
,
2
and vice versa.
The maximum
C
1
,
2
∼
10
2
that were obtained suggest
that internal conversion efficiencies
|
S
ij
|
2
η
1
η
2
∼
1 can be
achieved.
3.4. Sideband cooling
To estimate the noise generated by this device we
perform motional sideband cooling [69, 70, 71, 72, 27]
using each resonator mode independently, i.e. with only
one cooling tone at a time. The data is obtained
by measuring the electronic noise spectrum due to
the thermal Brownian motion of the nanobeam at the
resonance frequency
ω
m
with the linewidth Γ = Γ
i
+
γ
m
as a function of the cooling drive tone power
P
d
,i
applied at the optimal detuning ∆
i
=
ω
i
−
ω
m
. The
data shown in Fig. 2(e) and f for modes 1 and 2 has
been calibrated and analyzed as described in Ref. [27].
For small drive photon numbers we see a decrease of
the phonon occupancies
n
m
,i
(blue squares) in line with
the expectations (blue lines), i.e.
n
m
,i
=
n
b
C
i
+1
with
n
b
= 60 the mechanical bath corresponding to the
refrigerator temperature of
∼
12 mK and the intrinsic
mechanical linewidth
γ
m
(blue squares).
However,
we also observe a power dependent increase of the
noise floor with the bandwidth of the resonator mode
[33]. Using a theoretical model [73, 27] we fit the
resonator occupations
n
r
,i
, which we show (red circles)
together with a power law fit (dashed red line) in
Fig. 2(e) and (f). The power dependent resonator noise
limits the minimum phonon occupation to
n
m
∼
5 at
drive powers of about -5 dBm, independent of which
resonator mode is used for cooling.
The residual thermal population of the mechanical
oscillator (
n
m
∼
5) and the two microwave resonators
(
n
r
,i
∼
4) leads to incoherent added noise when the
device is used as a transducer. In the limit
C
1
∼
C
2
1 and the realistic assumption that the waveguide
modes are well thermalized and unpopulated, the
noise added to any converted signal at the output of
resonator port
i
is given as [45]
n
add
,i
=
η
i
(
n
r
,
1
+
n
r
,
2
+ 2
n
m
)
,
(6)
which results in
n
add
,i
∼{
16
,
12
}
photons
·
s
−
1
·
Hz
−
1
.
For this estimate we have cautiously assumed the same
mechanical population as measured with only a single
drive tone, due to the cooling limitations imposed by
the drive dependent resonator noise. In addition, finite
sideband resolution could lead to gain and amplified
vacuum noise. With sideband resolution factors of
κ
i
4
ω
m
≤
10
−
3
this is expected to be negligible in the
presented devices.
4. Wavelength conversion
4.1. Scattering parameters
In order to measure and quantify the efficiency of
the wavelength converter we obtain the full set of
scattering parameters for all 56 reported cooperativity
combinations.
The two reflection coefficients
|
S
ii
|
2
are extracted from the center region of the two-
mode EIT response shown in Fig. 2(b).
For the
transmission we apply a coherent signal on resonance
with one of the two microwave resonators and measure
the mechanically transduced signal appearing in the
center of the other resonator using a spectrum
analyzer. Measuring this in both directions allows
us to calibrate the product
|
S
21
||
S
12
|
=
|
T
|
2
, i.e.
the total bidirectional conversion efficiency [34], using
the product of both measured off-resonant reflection
coefficients
√
α
1
β
1
√
α
2
β
2
→
1, which were already
used to normalize the reflection parameters in Fig. 2(a)
and (b). In practice, we sweep the signal frequency in
a small range
δ
with a span on the order of the damped
mechanical linewidth and extract the maximum value
of the conversion efficiency. The result is shown in
Fig. 3(a) as a function of both cooperativities where red
color indicates high and blue color indicates low values
of the three scattering parameters. Qualitatively we
see that matching the cooperativities leads to higher
transmission and minimizes the reflection.
In panel (b) of Fig. 3 we show the quantitative re-
sult for the three cooperativity combinations indicated
with white lines in panel (a). In the first case we show
all three
S
parameters as a function of
C
1
for
C
2
≈
30.
As expected we observe the maximum conversion effi-
ciency of
|
S
ij
|
2
and the minimum reflection
|
S
ii
|
2
for
C
1
∼
C
2
. In the second plot we fix a higher value
(
C
1
≈
95) where we find that the optimal matching
condition is relaxed, i.e. the high conversion efficiency
is achieved for a larger range of
C
2
. Finally, keeping the
product constant
C
1
·
C
2
≈
660 and plotting the scat-
tering parameters as a function of the ratio
C
1
/C
2
, the
matching condition
C
1
=
C
2
is very clear. Although
the cooperativities are highest in the second case, the
maximum conversion efficiency of
|
S
ij
|
2
≈
0
.
6 is the
same due to the predicted limitations imposed by the
finite waveguide coupling efficiencies of
η
1
η
2
≈
0
.
63.
The estimated internal photon to photon conversion
efficiency is
|
S
ij
|
2
η
1
η
2
≈
0
.
95. These results are in excellent
agreement with the simple theory presented in Eq. (2)