Research Article
Vol. 31, No. 14 /3 Jul 2023 /
Optics Express
22914
Design of an ultra-low mode volume
piezo-optomechanical quantum transducer
P
IERO
C
HIAPPINA
,
1,2
J
ASH
B
ANKER
,
1,2
S
RUJAN
M
EESALA
,
1,2
D
AVID
L
AKE
,
1,2
S
TEVEN
W
OOD
,
1,2
AND
O
SKAR
P
AINTER
1,2,3,*
1
Thomas J. Watson Sr. Laboratory of Applied Physics, California Institute of Technology, Pasadena, CA
91125, USA
2
Kavli Nanoscience Institute and Institute for Quantum Information and Matter, California Institute of
Technology, Pasadena, CA 91125, USA
3
AWS Center for Quantum Computing, Pasadena, CA 91125, USA
*
opainter@caltech.edu
Abstract:
Coherent transduction of quantum states from the microwave to the optical domain
can play a key role in quantum networking and distributed quantum computing. We present the
design of a piezo-optomechanical device formed in a hybrid lithium niobate on silicon platform,
that is suitable for microwave-to-optical quantum transduction. Our design is based on acoustic
hybridization of an ultra-low mode volume piezoacoustic cavity with an optomechanical crystal
cavity. The strong piezoelectric nature of lithium niobate allows us to mediate transduction via
an acoustic mode which only minimally interacts with the lithium niobate, and is predominantly
silicon-like, with very low electrical and acoustic loss. We estimate that this transducer can
realize an intrinsic conversion efficiency of up to 35% with
<
0.5 added noise quanta when
resonantly coupled to a superconducting transmon qubit and operated in pulsed mode at 10 kHz
repetition rate. The performance improvement gained in such hybrid lithium niobate-silicon
transducers make them suitable for heralded entanglement of qubits between superconducting
quantum processors connected by optical fiber links.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Recent landmark demonstrations have established superconducting quantum circuits as a leading
platform for quantum computing [1–4]. Superconducting processors encode quantum information
in microwave-frequency photons and require operation at milliKelvin temperatures. Due to the
high propagation loss and thermal noise of microwave links at room temperature, transmitting
quantum information from superconducting processors over long distances remains an outstanding
challenge. In contrast, optical photons are naturally suited for low loss, long distance transmission
of quantum information [5,6]. The complementary properties of these two systems have spurred
interest in transducers that can coherently convert quantum information between microwave and
optical frequencies. Such transducers would enable optically connected networks of remote
superconducting quantum processors analogous to classical networks underlying the internet and
large-scale supercomputers with optical interconnects.
A quantum transducer operated as a frequency converter can be specified as a linear device
with a certain conversion efficiency, added noise level, and repetition rate for pulsed operation.
Current approaches for microwave-to-optical frequency conversion rely on a strong optical pump
to mediate the conversion process between single photon-level signals at both frequencies [7–22].
Increasing pump power allows for higher conversion efficiency, but due to parasitic effects of
optical absorption in various components of the transducer, and the vast difference in energy
scales between optical and microwave frequencies, this adds noise to the conversion process. For
quantum applications of the transducer, the number of added noise photons per up-converted
#493532
https://doi.org/10.1364/OE.493532
Journal © 2023
Received 18 Apr 2023; revised 7 Jun 2023; accepted 13 Jun 2023; published 23 Jun 2023
Research Article
Vol. 31, No. 14 /3 Jul 2023 /
Optics Express
22915
signal photon should be
≲
1 [23]. This trade-off between efficiency and noise has been a key
obstacle to transduction of quantum signals.
One particularly promising approach to microwave-to-optical transduction is piezo-optomechanics
[12–18], in which acoustic phonons are used as intermediate excitations in the conversion process.
This is achieved through a highly engineered mechanical mode with simultaneous piezoelectric
and optomechanical couplings. Recent design and materials advances in these devices have led
to a demonstration of optical readout of the quantum state of a superconducting qubit with added
noise below one photon [12]. However, the aluminum nitride piezoelectric element contributed
significantly to acoustic loss, compromising device performance.
In this work, we propose a new device design on a hybrid material platform, lithium niobate
(LN) on silicon, which addresses the limitations of previous work. Our design features a
highly miniaturized piezoelectric element so that its negative impact on device performance is
negligible, while at the same time maintaining strong piezoelectric coupling rates. This design
approach is enabled by the strong piezoelectric coefficients of LN [24]. Combined with the
excellent optomechanical properties of silicon, our design achieves state-of-the-art piezoelectric
and optomechanical performance. While some recent demonstrations have used a lithium
niobate-on-silicon material platform [14,17], the design techniques outlined in this work were
not fully exploited in these devices. We show that by leveraging the strengths of the individual
materials, our design approach can yield the performance improvements necessary to employ
these transducers for remote entanglement of superconducting quantum processors.
Figure 1(a) illustrates the mode picture of our transduction scheme. An intermediary mode
ˆ
b
m
of a nanomechanical oscillator simultaneously couples to microwave photons from mode
ˆ
c
q
of a microwave circuit at rate
g
pe
, and to optical photons from mode
ˆ
a
o
of an optical
cavity at rate
g
om
. Microwave photons are converted to phonons via a resonant piezoelectric
interaction, and these phonons are subsequently converted into optical photons via a parametric
optomechanical interaction. The microwave photon-phonon conversion is realized by tuning the
circuit frequency
ω
q
on resonance with the mechanical frequency
ω
m
, yielding the Hamiltonian
ˆ
H
pe
/
ℏ
=
g
pe
(
ˆ
b
†
m
ˆ
c
q
+
ˆ
b
m
ˆ
c
†
q
)
[25]. The phonon-optical photon conversion is realized by driving
the optical cavity at a frequency
ω
d
that is red-detuned by exactly the mechanical frequency,
ω
d
−
ω
o
=
−
ω
m
. The resulting Hamiltonian is
ˆ
H
om
/
ℏ
=
g
om
√
n
o
(
ˆ
a
†
o
ˆ
b
m
+
ˆ
a
o
ˆ
b
†
m
)
where
n
o
is the
number of intracavity optical photons corresponding to input drive power [26].
We realize the intermediary mechanical mode in the above scheme by connecting an ultra-low
mode volume piezoacoustic cavity and an optomechanical crystal (OMC) cavity, shown in
Fig. 1(b). The acoustic modes of these components are strongly hybridized to form a mechanical
supermode, whose mechanical displacement overlaps in one region with the field of a microwave
circuit, and in another region with the field of an optical cavity. Using physically separate cavities
allows us to independently optimize the piezoacoustic and optomechanical components of the
transducer. Our design is formed from thin-film LN on the device layer of a silicon-on-insulator
(SOI) chip. We define the piezoacoustic cavity in LN, which has large piezoelectric coefficients
[24]. We define the OMC in silicon, since its large photoelastic coefficients and refractive index
allow high optomechanical coupling. Well-established nanofabrication processes also allow high
optical and mechanical quality factors for silicon OMC’s [27–29]. For the microwave circuit in
this design, we consider a transmon qubit with electrodes routed over the LN region to allow
for capacitive coupling to the piezoacoustic cavity. The transmon is patterned in niobium on
silicon, a standard material platform for realizing high-coherence qubits [30]. The insulating
layer underneath these components is etched away, leaving a suspended silicon membrane as the
substrate for our device.
Our design procedure utilizes finite element simulation in COMSOL Multiphysics and
numerical optimization of the device geometry. We begin with independently optimizing the
piezoacoustic and OMC cavities for high piezoelectric coupling rate
g
pe
and optomechanical
Research Article
Vol. 31, No. 14 /3 Jul 2023 /
Optics Express
22916
Piezoacoustic
Cavity
Optomechanical
Cavity
iez
oa
c
ou
stic
C
avit
y
Opt
omechanic
al
C
avit
y
5 GHz
5 GHz
194 THz
Fig. 1.
a) Mode schematic for piezo-optomechanical transduction. Blue represents mode
ˆ
c
q
of a microwave circuit, purple represents ’supermode’
ˆ
b
m
of a mechanical oscillator, and red
represents mode
ˆ
a
o
of an optical cavity. b) Device schematic for the transducer in this work.
The device can be split into two regions, one which couples to microwave electric fields and
one which couples to optical fields. Both are part of the same mechanical ’supermode’
ˆ
b
m
.
coupling rate
g
om
, respectively. We design for closely matched acoustic modes at 5 GHz in both
resonators, and for an optical mode at telecom wavelength (1550 nm). During the design process,
it is crucial to maintain a low acoustic mode density such that the transduction schematic in
Fig. 1(a) using a single acoustic mode remains valid. Further, since thin film LN has higher
microwave dielectric and acoustic loss than silicon [29,31–34], we aim to minimize the volume
of LN in our device. The two independently optimized cavities are then physically connected
and the parameters of the resulting hybrid acoustic modes are analyzed.
2. Piezo cavity design
The piezoacoustic cavity consists of a slab of lithium niobate on top of a suspended silicon
membrane patterned in the shape of a box. We work with 100 nm thin-film -z-cut LN, on top
of a 220 nm-thick suspended silicon device layer. We choose the -z LN orientation so that the
dominant LN piezoelectric coefficient,
d
15
=
69 pC/N [24], couples to modes with favorable
symmetry properties (further discussion in Sec. 3.). 80 nm-thick Nb electrodes run over the
top of the slab, and are routed in the form of an interdigital transducer (IDT) which capacitively
couples the cavity to a microwave circuit such as a transmon qubit, as seen in Fig. 2(a).
The box is surrounded by a periodically patterned phononic shield to mitigate acoustic
radiation losses and to clamp the membrane to the surrounding substrate. The clamps are spaced
periodically so that the IDT electrodes are routed over the top of each clamp, providing a means
for electrical routing which is not acoustically lossy.
The phononic shield uses an alternating block and tether pattern (see relevant dimensions
in Fig. 2(c)) consisting of metal electrodes on top of a silicon base. By tuning the parameters
a
,
b
x
,
b
y
, and
t
y
, we achieve a
>
1 GHz acoustic bandgap centered around 5.1 GHz, shown in
Fig. 2(d). This yields strong confinement of 5 GHz mechanical modes inside the piezo region for
Research Article
Vol. 31, No. 14 /3 Jul 2023 /
Optics Express
22917
Frequency (GHz)
Silicon
Piezo Box
Electrodes
k(π/a)
0
1
8
6
4
2
log
10
(u
m
/max(u
m
))
-5
0
x
y
z
Fig. 2.
a) Schematic illustrating the capacitive routing of the piezoacoustic cavity to a
transmon qubit. The qubit here can be replaced with a microwave resonator without loss of
generality. b) Piezoacoustic cavity geometry, with relevant dimensions defined in blowout
top view. c) Phononic shield unit cell, with relevant dimensions defined. d) Mechanical
bandstructure of phononic shield unit cell in c), with
(
a
,
b
x
,
b
y
,
t
y
)
=
(
445, 225, 265, 70
)
nm.
Black dashed line at 5.1 GHz shows the target frequency of the mode of interest. We observe
a complete acoustic bandgap in excess of 1 GHz around 5.1 GHz. e) Log scale of mechanical
energy density
u
m
for piezoacoustic cavity mode at 5.1 GHz, normalized to maximum value.
We find
>
4 orders of magnitude suppression for 5 phononic shield periods.
sufficient number of shield periods, enabling high mechanical quality factors. By simulating
the mechanical energy density across the entire cavity, we find that 5 shield periods provide
>
4
orders of magnitude suppression of acoustic radiation into the environment, as shown in Fig. 2(e).
The dimensions of the piezo box (outlined in Fig. 2(b)) are designed to support a periodic
mechanical mode whose periodicity matches that of the IDT fingers. This results in high overlap
between the electric field from the IDT and the electric field induced by mechanical motion in
the piezo box. This overlap gives a microwave photon-phonon piezoelectric coupling rate which
is derived using first order perturbation theory:
g
pe
=
ω
m
4
√︁
2
U
m
U
q
∫
LN
D
m
·
E
q
dV
.
(1)
Here the integral is taken over the entire LN slab,
D
m
is the electric displacement field induced
from mechanical motion in the piezo region, and
E
q
is the single-photon electric field generated
by the transmon qubit across the IDT electrodes. The fields are normalized to their respective
zero-point energies
ℏ
ω
m
/
2, yielding the pre-factor in front of the integral in Eq. (1).
U
m
is the
total cavity mechanical energy, and
U
q
=
1
2
(
C
q
+
C
IDT
)
V
2
0
is the total IDT electrostatic energy,
where
V
0
is the zero-point voltage across the qubit. We note that for fixed electrostatic energy, the
zero-point voltage (and thus
E
q
) is dependent on both the qubit capacitance
C
q
and IDT finger
capacitance
C
IDT
, and therefore the coupling rate scales as
(
C
q
+
C
IDT
)
−
1
/
2
. For our calculations
in this work, we assume
C
q
=
70 fF which is a typical value for transmon qubit capacitance
[35]. Replacing the transmon qubit with a high-impedance microwave resonator will allow lower
Research Article
Vol. 31, No. 14 /3 Jul 2023 /
Optics Express
22918
C
q
∼
2 fF [36,37], and can therefore further increase this coupling rate.
C
IDT
is calculated with
finite-element electrostatic simulation, and for the wavelength-scale devices considered here is
typically
∼
0.25 fF, a small contribution compared to transmon
C
q
.
The small value of
C
IDT
also minimizes the energy participation of the qubit electric field
in the lossy piezo region, given by the ratio
ζ
q
=
C
IDT
/
C
q
∼
4
×
10
−
3
. The contribution
of lithium niobate to the qubit loss rate
κ
q
,
i
is then estimated as
ζ
q
κ
q
,
LN
. Using reported
dielectric loss tangents in lithium niobate at milliKelvin temperatures,
tan
δ
=
1.7
×
10
−
5
[32,33]
corresponding to
κ
q
,
LN
/
2
π
=
85 kHz at 5GHz, we estimate the LN contribution to qubit loss to be
ζ
q
κ
q
,
LN
/
2
π
∼
300 Hz. This contribution is much smaller than typical loss rates
κ
q
,
SOI
/
2
π
∼
50
kHz reported in transmon qubits fabricated on SOI [38]. As a result, the contribution of the
piezoacoustic cavity to qubit loss is not a limiting factor, and justifies the on-chip coupling
scheme outlined in Fig. 2(a).
We note that the IDT capacitance can be increased by adding more IDT fingers, thereby
increasing
g
pe
. However, as the number of fingers increases, the increased cavity size results
in a more crowded mode structure, and it is more difficult to isolate a single mechanical mode
without coupling to parasitic modes in the vicinity of the mode of interest. This is of key
importance as these parasitic modes may not hybridize well with the OMC cavity and reduce
overall transduction efficiency. This is shown in Fig. 3(b), where we simulate the mechanical
mode structure vs. the number of IDT fingers in the cavity. We see that
g
pe
of the mode of
interest saturates, while the mode isolation is significantly reduced with increasing IDT fingers.
IDT Fingers
Mode Isolation (MHz)
Frequency (GHz)
1
-1
1
-1
Frequency (GHz)
Period (nm)
z Displacement
y Displacement
0
5
10
4.9
5.0
5.1
5.2
5.3
g
pe
/2π (MHz)
g
pe
/2π (MHz)
5
10
15
20
g
pe
/2π (MHz)
x
y
2
4
6810
0
50
100
150
0
5.3
5.2
5.1
5.0
4.9
10
700
750
800
850
900
8
6
4
2
5.4
Fig. 3.
a) Mode structure for optimized piezoacoustic cavity design with
(
p
,
w
p
,
e
)
=
(
795, 478, 102
)
nm. Mode of interest (red) achieves
g
pe
/
2
π
of 9 MHz. Shaded grey region
indicates the mode isolation window with the nearest mode
>
110 MHz away. b)
g
pe
/
2
π
and
mode isolation for optimized designs with differing number of IDT fingers. We observe
g
pe
saturating beyond
N
=
4 fingers, and mode isolation decreasing with increasing number of
fingers. c) Mechanical mode shape of red mode in a). Right shows the in-plane (breathing)
and left shows the out-of-plane (Lamb-wave) components of the optimized mechanical mode.
d) Mode structure vs. IDT period, with data points colored according to
g
pe
. Dashed line
follows the mode of interest.