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Efficient microwave frequency conversion mediated by a photonics
compatible silicon nitride nanobeam oscillator
To cite this article: J M Fink
et al
2020
Quantum Sci. Technol.
5
034011
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Quantum Sci. Technol.
5
(2020) 034011
https://doi.org/10.1088/2058-9565/ab8dce
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PAPER
Efficient microwave frequency conversion mediated by a
photonics compatible silicon nitride nanobeam oscillator
JMFink
1
,
2
,
3
,
6
,MKalaee
1
,
2
,RNorte
1
,
2
,
4
,APitanti
1
,
2
,
5
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, United States of America
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, United States of America
3
Institute of Science and Technology Austria (IST Austria), 3400 Klosterneuburg, Austria
4
Present address: Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628CD Delft,
The Netherlands.
5
Present address: NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, Piazza San Silvestro 12, 56127 Pisa (PI), Italy.
6
Author to whom any correspondence should be addressed.
E-mail:
jfink@ist.ac.at
Keywords:
superconducting circuits, electromechanics, optomechanics, MEMS, frequency conversion, hybrid devices, silicon nitride
membranes
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 realiz
e ultra-sensitive, low-loss transducers for an
emerging quantum information technology. Her
e we present an on-chip microwave frequency
converter based on a planar aluminum on silic
on 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 bidir
ectional 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.
1. Introduction
Silicon nitride (Si
3
N
4
) thin films show exceptional optical and mechanical properties [
1
], and are used in
many microelectromechanical and photoni
c devices. The material’s large bandgap [
2
], high power handling
due to the absence of two-photon ab
sorption 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
]andformupto
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
–
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 phot
onic crystal cavities have been
developed using Si
3
N
4
thin films to realize strong optomechanica
l interactions in a small mode volume and
fully integrated on-chip [
20
–
23
].
In the microwave domain, Si
3
N
4
is widely used for wiring capacitors and cross-overs. Early work
focused on the study of Si
3
N
4
as a low loss dielectric to realize compac
t 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 di
pole moments and life-times, which led to the
© 2020 IOP Publishing Ltd
Quantum Sci. Technol.
5
(2020) 034011
JMFink
et al
observation of strong coupling between a single two-level system and a superconducting resonator [
26
–
28
].
Nonetheless, due to its unique mechanical properties, high quality factor membranes [
29
,
30
]aswellas
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 participati
on 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 capacitan
ce 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 demo
nstrating sideband cooling of the low MHz frequency
nanobeam to the motional ground state [
27
].
Silicon nitride membrane-based devices are currently the leading approach to couple optical and
microwave systems [
34
,
35
]. Realizing noiseless conversion with a mechanical oscillator [
36
–
43
] would
allow one to build transducers for quantum networks of superconducting processors connected via
resilient and low loss optical fiber networks [
44
–
46
]. Efficient wavelength conversion has been realized
between optical wavelengths using s
ilicon optomechanical crystals [
47
], between microwave frequencies
using metallic drum resonators [
48
–
50
] and silicon nanobeams [
51
], and also between microwave and
optical wavelength using silicon nitride
membrane based Fabry–Perot cavities [
34
,
35
] and very recently
with silicon nanobeam photonic crystals [
52
]. Alternative approaches include the use of Josephson circuits
for conversion in the microwave domain [
53
–
56
], 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 [
57
,
58
], via a mechanical intermediary in combination
with the piezoelectric effect and optomechanical interactions [
59
–
62
], and microwave to optics conversion
has been proposed [
63
–
65
] and realized with high bandwidth via the electro-optic effect [
66
,
67
].
At room temperature wide-band microwave frequency
conversion is done using commercial diode based
mixers with a typical conversion loss of 6 dB. At low temperatures microwave mixing has been realized
with Josephson circuits [
54
–
56
] for quantum level signals with up to 10’s of MHz bandwidth and a
conversion loss as low as 0.05 dB. Using parametrically driven mechanical systems for such as task typically
results in lower bandwidth and potentially more noise,
in particular if a low frequency intermediary is used.
However, compared to Josephson circuits, mechanical transducers show orders of magnitude higher
dynamic range in a more compact design with little cross-talk between mechanical modes and better
resilience to stray magnetic fields. Reference [
53
] demonstrated mechanically mediated microwave
conversion with an aluminum drum oscillator, achi
eving an efficiency of 95% with less than 0.1 added
conversion noise quanta and a dynamic range exceeding 10
12
photons per second over a bandwidth of
14 kHz. However, in the context of microwave to optics conversion it is not known how such metallic
oscillators or Josephson circuits could be ef
ficiently coupled to integrated photonics.
Dielectric mechanical systems offer the potentia
l 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. Our experiment is
based on such a dielectric nanobeam oscillator and sh
ares the advantages of mechanical systems compared
to Josephson devices. While we do not achieve the same figures of merit as in reference [
53
], our
experiment shows a clear path forward for coupling the dielectric mechanical intermediary to integrated
photonics. The presented microwave frequency converter on the aluminum-on-Si
3
N
4
platform [
27
], is
compatible with on-chip optomechanics designs worked out in references [
21
,
22
] and directly applicable to
the low frequency silicon design in reference [
52
]. Compared to the latter, we anticipate that similar or
improved coupling strength, transduction efficiencies and bandwidths are achievable, because the
structurally stable and fully under-etched membrane
design lowers the stray capacitances of the resonator
inductance further [
27
], the techniques of soft-clamping coul
d help to improve the mechanical quality
factor [
16
–
18
], and the demonstrated V-groove based packaging can maximize fiber to chip coupling
efficiencies [
11
]. However, high yield fabrication based on a wet release is challenging [
27
], the lower
refractive index and dielectric permittivity compared to silicon require even smaller capacitor and slot
mode gaps, and it is difficult to reliable assess the expected power handling, thermal conductivity and added
noise due to optical absorption heating in a new material. A reliable quantification and comparison will
therefore require careful tests at millikelvin temper
atures. Nevertheless, here we show that an approach
based on Si
3
N
4
is feasible and promising in order to realize on-chip conversion between microwave and
optical fields via radiation pressure, or to implement ultra-low voltage modulation and fully electrical
tunability in Si
3
N
4
-based photonic devices in the near future.
2
Quantum Sci. Technol.
5
(2020) 034011
JMFink
et al
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 references [
37
–
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 the mechanical and the two
electromagnetic modes, see figure
1
(a). In the resolved-sideband limit (
ω
m
κ
i
,
γ
m
)thelinearized
electromechanical Hamiltonian in the rotating frames and the rotating wave approximation is given by
H
=
∑
i
=
1,2
Δ
i
ˆ
a
†
i
ˆ
a
i
+
ω
m
ˆ
b
†
ˆ
b
+
∑
i
=
1,2
g
i
(
ˆ
a
i
ˆ
b
†
+
ˆ
b
ˆ
a
†
i
)
,(1)
where
â
i
is the annihilation operator for the microwave field mode,
ˆ
b
is the annihilation operator of the
mechanical mode,
Δ
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
ω
d,
i
4
κ
ex,
i
κ
2
i
+
4
Δ
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 equation (
1
) have two closely related effects. Optomechanical
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
theory, 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
â
in,1
is down-converted to the mechanical mode at frequency
ω
m
, which corresponds to
ˆ
b
†
ˆ
a
1
in equation (
1
). Next, during an up-conversion process the mechanical mode transfers its energy to the
output of the other microwave resonator at frequency
ω
2
and amplitude
â
out,2
, which corresponds to
ˆ
a
†
2
ˆ
b
in
equation (
1
). In this process the mechanical resonance is virtually populated, in the sense that the input
signal is rapidly converted to the output 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 figure
1
(a). The Hermitian aspect of the Hamiltonian (
1
)makesthis
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
.Bysolvingthelinearized
Langevin equations we find that for signals on resonanc
e with the microwave resonator, in the steady state
the bidirectional conversion efficiency is given as [
68
]
|
S
ij
|
2
=
|
S
ji
|
2
=
|
T
|
2
=
η
i
η
j
4
C
i
C
j
(1
+
C
i
+
C
j
)
2
,(2)
for
i
=
j
with
i
,
j
=
{
1, 2
}
the indices of the two modes.
C
i
=
Γ
i
γ
m
is the electromechanical cooperativity for
resonator
i
and
η
i
=
κ
ex,
i
κ
i
is the resonator coupling efficiency 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 given by
|
S
ii
|
2
=
(
1
−
2
η
i
(1
+
C
j
)
1
+
C
i
+
C
j
)
2
(3)
for
i
,
j
=
{
1, 2
}
and
i
=
j
. For lossless microwave cavities
η
i
=
1 and in the limit
C
1
=
C
2
1nearunity
photon conversion efficiency with
|
T
|
2
=
1and
|
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
,whilethelatter
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-pho
ton 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.
3
Quantum Sci. Technol.
5
(2020) 034011
JMFink
et al
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
}
,aswellastheconversionscatteringparameters
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) i
s coupled capacitively via its two modulated capacitances
C
m,
i
to two
parallel inductance–capacitance resonators realized with hig
h characteristic impedance pl
anar 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
â
in,
i
and
â
out,
i
. (c) False color scanning electron micrograph of the converter device
with thin-film aluminum (white) on suspended silicon nitride
membrane (blue). Mechanical b
eam, 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 semi-rigid 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 K stage
and also at room temperature before detection with either a spectrum analyzer (SA) or the VNA input.
2.2. Circuit
We implement bidirectional frequency conversion in a circuit as shown in figure
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 forme
d from a planar spiral inductor of high impedance.
The capacitance of these lumped element resonant circ
uits 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 two inductors.
2.3. Device
The described circuit is fabricated on the aluminum-on-Si
3
N
4
platform similar to reference [
27
]. Here the
entire aluminum circuit, which is shown in figure
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
4