Quantum electromechanics of a hypersonic crystal
Mahmoud Kalaee,
1, 2,
∗
Mohammad Mirhosseini,
1, 2,
∗
Paul B. Dieterle,
1, 2
Matilda Peruzzo,
3
Johannes M. Fink,
3, 2
and Oskar 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, 3400, Klosterneuburg, Austria
(Dated: September 3, 2018)
Radiation pressure within engineered structures has recently been used to couple the motion of
nanomechanical objects with high sensitivity to optical and microwave electromagnetic fields. Here,
we demonstrate a form of electromechanical crystal for coupling microwave photons and hypersonic
phonons by embedding the vacuum-gap capacitor of a superconducting resonator within a phononic
crystal acoustic cavity. Utilizing a two-photon resonance condition for efficient microwave pumping
and a phononic bandgap shield to eliminate acoustic radiation, we demonstrate large cooperative
coupling (
C
≈
30) between a pair of electrical resonances at
ω
r,
0
≈
10 GHz and an acoustic resonance
at
ω
m
/
2
π
= 0
.
425 GHz. Electrical read-out of the phonon occupancy shows that the hypersonic
acoustic mode has an intrinsic energy decay time of 2
.
3 ms and thermalizes close to its quantum
ground-state of motion (occupancy
n
m
= 1
.
5) at a fridge temperature of
T
f
= 10 mK. Such an
electromechanical transducer is envisioned as part of a hybrid quantum circuit architecture, capable
of interfacing to both superconducting qubits and optical photons.
Superconducting electromechanical circuits have re-
cently been used to demonstrate exquisitely sensitive
measurement and control of mesoscopic mechanical ob-
jects in the quantum regime [1–3]. In these devices, me-
chanical coupling is typically via a dispersive interaction
with a high
Q
-factor electromagnetic resonator, whereby
mechanical motion phase modulates the internal electro-
magnetic field resulting in the generation of motional
sidebands. This interaction is parametric, allowing for
the coupling of low frequency mechanical oscillators to
much higher frequency electromagnetic resonators, and
can be enhanced through application of a strong elec-
trical driving tone. The strength of the coupling at the
quantum level is defined by a vacuum rate,
g
0
, which
for capacitively coupled circuits is related to the scale of
the mechanical quantum zero-point motion in compari-
son to the dimension of the capacitor. Superconducting
microwave resonators employing nanoscale vacuum-gap
capacitors can reach vacuum coupling levels as large as
a few hundred Hz to MHz-frequency mechanical oscil-
lators [4], and have been utilized for a variety of appli-
cations ranging from conversion between microwave and
optical photons [5, 6] to the generation and detection of
squeezed states of mechanical motion [7–9].
Similar work in the optical domain has sought to in-
crease the radiation pressure within optical resonators
by scaling the optical mode volume down to the
nanoscale [10]. An example of this is the optomechan-
ical crystal (OMC) [11], in which large coupling be-
tween near-infrared photons and hypersonic (
>
∼
GHz) [12]
phonons has been realized.
Owing to the factor of
∗
These authors contributed equally to this work.
†
opainter@caltech.edu
∼
10
5
between the speed of light and sound in solid-
state materials, optical photons and hypersonic phonons
are matched in wavelength, enabling the construction of
integrated photonic and phononic circuits which can be
used to route signals around on a chip or to inter-convert
optical and acoustic waves [13–16]. For quantum appli-
cations, hypersonic acoustic devices also have several ad-
vantages. GHz-level frequencies facilitate operation in
the sideband-resolved limit of optomechanics [10], a cru-
cial parameter regime for realizing noise-free quantum
signal conversion [5, 17]. Additionally, the integration
of superconducting quantum circuits [18] and microwave
acoustic devices is actively being explored [3, 19–26],
where the compact wavelength and lack of fringing fields
in vacuum of acoustic phonons can enable superior minia-
turization and scaling.
Towards this effort, the use of piezoelectric materi-
als [27, 28] can enable MHz-rate electromechanical cou-
pling suitable for quantum information processing [13,
22–26, 29, 30].
The piezoelectric coupling, however,
cannot be turned off nor is it perfectly mode selective,
and poly-crystalline piezoelectric materials can harbor
lossy defects [31]. Both these effects can lead to para-
sitic electrical or acoustic decoherence. Parametric ra-
diation pressure coupling can be dynamically controlled
and is relatively materials agnostic, but it is challeng-
ing to reach the requisite level of coupling due to the
large mismatch in electromagnetic and acoustic wave-
lengths. Using an aluminum (Al) on silicon-on-insulator
(SOI) process which has been effective in forming low-loss
superconducting quantum circuits [32], here we demon-
strate an electromechanical resonator that utilizes hyper-
sound frequency phononic crystals to engineer the local-
ization and parametric coupling of mechanical motion at
ω
m
/
2
π
= 0
.
425 GHz to an integrated superconducting
arXiv:1808.04874v2 [quant-ph] 30 Aug 2018
2
microwave high impedance circuit. This electromechani-
cal crystal (EMC) structure, akin to the optical OMCs,
achieves simultaneously the large photon-phonon cou-
pling (
g
0
/
2
π
= 17
.
3 Hz) and minimal acoustic damping
(
γ
i
/
2
π
= 68 Hz) required of quantum electromechanics
applications.
Electromechanical crystal design and fabrication
The electromechanical crystal studied in this work is
formed from superconducting Al wiring on a patterned
sub-micron thick silicon (Si) membrane, and consists of
three primary elements: (i) a central nanobeam phononic
crystal cavity and capacitor with nanoscale vacuum gap,
(ii) an acoustic shield with a phononic bandgap for all po-
larizations and propagation directions of acoustic waves,
and (iii) a nanoscale-pitch spiral coil inductor with min-
imal stray capacitance and large intrinsic impedance.
Details of the planar spiral inductor are described in
App. A. Here we focus on the design of the nanobeam
cavity and acoustic shield. Figure 1(a) depicts the pat-
terned nanobeam cavity geometry and Al wiring layout
of the vacuum-gap capacitor. The resulting hypersonic
‘breathing’ acoustic cavity mode is also shown, visualized
as an exaggerated deformation of the beam structure.
Referring to the nanobeam unit cell and acoustic band-
structure of Figs. 1(b-c), this breathing mode is formed
from an acoustic band (solid bold red curve) near the
Γ-point at wavevector
k
x
= 0. For a lattice constant of
a
n
= 1
.
55
μ
m numerical finite-element method (FEM)
simulations place the Γ-point frequency of the breath-
ing mode band at
ω
m
/
2
π
= 0
.
425 GHz. Although other
acoustic bands (dashed curves) are also present, the rel-
ative isolation of the breathing Γ-point modes in recipro-
cal space still allows for the formation of highly localized
cavity modes near the band-edge.
Several subtle features of the nanobeam design are key
to realizing large electromechanical coupling, the magni-
tude of which is given by [33],
g
0
=
−
ηx
zpf
ω
r,
0
2
C
m
∂C
m
∂u
,
(1)
where
ω
r,
0
is the resonance frequency of the coupled LC
circuit,
u
is the displacement amplitude of the acoustic
mode of interest,
x
zpf
is the zero-point amplitude of this
mode, and
C
m
is the vacuum-gap capacitance affected
by the beams motion.
η
is a motional participation ra-
tio defined by
η
=
C
m
/C
tot
, where
C
s
is the stray and
C
tot
=
C
s
+
C
m
the total capacitance of the LC circuit.
Firstly, a minimum motional mass (
m
eff
) is desired to in-
crease the zero-point amplitude (
x
zpf
= [ ̄
h/
2
m
eff
ω
m
]
1
/
2
).
In the case of the patterned nanobeam this corresponds
to minimizing the thickness of the Si and Al layers and
minimizing the width of the beam features. Secondly,
a large motional capacitance is desired due to limits on
the achievable stray capacitance. Owing to the use of
a Γ-point acoustic mode the electromechanical coupling
from each unit cell is additive and increasing the number
of unit cells in the acoustic cavity results in an increased
w
n
w
x
a
n
w
y
b
c
d
0
100
200
300
400
500
600
frequency (MHz)
k
x
(
π/
a
n
)
0
1
35
45
55
65
75
1.4
1.6
1.8
2.0
12
20
28
36
gap size (nm)
C
m
(fF)
g
0
/2
¼
(Hz)
́
4
6
8
10
12
14
0.1
0.2
0.4
0.3
2
4
6
8
10
12
14
# of unit cells
e
a
C
s
L
x
zpf
(fm)
x
y
z
FIG. 1.
Nanobeam phononic crystal design. a
,
Schematic of the central nanobeam region showing the breath-
ing mode. Mechanical motion is indicated by an exaggerated
displacement of the beam structure, with red (blue) color in-
dicating regions of large (small) amplitude of the motion. The
Al capacitor electrodes (grey) are connected in parallel to a
coil inductor of inductance
L
and parasitic capacitance
C
s
.
b
, Unit cell of the nanobeam phononic crystal lattice with Si
device layer (Al electrode) shown as blue (grey).
c
, Acoustic
band structure for an infinitely periodic nanobeam phononic
crystal with parameters:
a
n
= 1
.
55
μ
m,
W
n
= 900 nm,
W
x
= 600 nm and
W
y
= 1
.
45
μ
m. The Si device layer and Al
electrode thicknesses are
t
Si
= 220 nm and
t
Al
= 60 nm, re-
spectively. The red and blue curves correspond to symmetric
and anti-symmetric modes with respect to the
x
-
z
symmetry
plane. The band from which the breathing mode is formed
is shown as a solid red curve.
d
, Participation ratio (
η
) and
zero-point motion amplitude (
x
zpf
) of the breathing mode as
a function of number of unit cells in the beam for a fixed para-
sitic capacitance
C
s
= 3
.
1 fF and a vacuum gap size of 45 nm.
e
, Motional capacitance,
C
m
, and zero-point coupling,
g
0
, of
the electrical circuit as a function of the gap size. Here the
coil inductance,
L
, is adjusted for each gap to keep the LC-
resonance frequency fixed at
ω
r,
0
= 10
.
77 GHz. See App. A
and B for details of mechanical and electrical numerical sim-
ulations.
motional capacitance and participation ratio. FEM sim-
ulations of
x
zpf
and
η
versus the number of nanobeam
unit cells are shown in Fig. 1(d) for a stray capacitance
C
s
= 3
.
1 fF and a fixed vacuum gap
s
= 45 nm. Here,
C
s
is dominated by the stray capacitance of the planar spi-
ral coil inductor forming the LC resonator. Figure 1(e)
shows the resulting simulated vacuum coupling rate ver-
sus gap size of the breathing mode for a nanobeam struc-
ture consisting of 11 unit cells. Beyond 11 unit cells we
find the acoustic mode becomes too sensitive to disorder,
3
log
10
(
ρ
/
ρ
0
)
0
-10
a
m
w
m
h
m
c
a
150
200
250
300
350
400
450
frequency (MHz)
in-plane
k
-vector
Γ
X
M
Γ
b
ω
m
FIG. 2.
Phononic crystal shield. a
, Schematic and di-
mensions of a unit cell in the phononic crystal shield (mirror
cell) surrounding the nanobeam central cavity.
b
, Acoustic
band structure of the acoustic shield for mirror unit cell pa-
rameters:
a
m
= 5
.
13
μ
m,
W
m
= 2
.
1
μ
m,
h
m
= 360 nm
and
t
Si
= 220 nm. The acoustic band gap is shaded in blue
and the localized breathing mode frequency is marked with a
dashed red line.
c
, Logarithmic scale color plot of the acoustic
energy density for the nanobeam breathing of (a) embedded
in the acoustic shield of (b). Acoustic energy density,
ρ
, is
normalized to its peak value,
ρ
0
, located in the nanobeam.
Displacement of the structure is also used to visualize the
breathing mode profile.
and tends to breaks up into localized resonances when
fabricated.
As mentioned above, the nanobeam phononic crys-
tal does not have a full phononic bandgap in the vicin-
ity of the breathing mode. In order to provide addi-
tional acoustic isolation from the surrounding Si mem-
brane and substrate the nanobeam cavity and vacuum-
gap capacitor are embedded in the middle of a ‘cross-
pattern’ phononic bandgap crystal [34]. A unit cell of
the cross shield, shown in Fig. 2(a), consists of a large
square plate region with four narrow connecting tethers.
Through adjustment in the width of the square plate,
and length and width of the connecting tethers, a sub-
stantial bandgap can be opened up between the low fre-
quency tether modes and the localized modes of each
square plate. A FEM-simulated acoustic bandstructure
of an optimized cross structure is shown in Fig. 2(b),
where a bandgap of nearly 0
.
1 GHz around the breathing
mode frequency of 0
.
425 GHz is obtained. Embedding
the nanobeam phononic crystal cavity in the middle of
a cross phononic crystal, Fig. 2(c) shows a simulation of
the resulting radiation pattern of the localized breathing
mode. As can be clearly seen, the energy density of the
breathing mode reduces dramatically upon entering the
acoustic shield, dropping by 100 dB in only two periods.
A final design consideration relates to the large hyper-
sound frequency of the breathing mode. Coupling this
mode to a microwave circuit of comparable frequency
introduces a large effective detuning in the parametric
interaction, greatly increasing the required microwave
pump power. We circumvent this problem by using a
multimode microwave cavity [35] consisting of two cou-
pled single-mode LC resonators. In this scheme, one
of the electromagnetic modes is resonant with the mi-
crowave pump tone, while the second mode is detuned
by the acoustic mode frequency. Formally, the coupling
of the acoustic mode to the multimode microwave cavity
can be described via the Hamiltonian
ˆ
H
= ̄
hω
r,
0
(ˆ
a
†
1
ˆ
a
1
+ ˆ
a
†
2
ˆ
a
2
) + ̄
hJ
(ˆ
a
1
ˆ
a
†
2
+ ˆ
a
†
1
ˆ
a
2
)
+ ̄
hω
m
ˆ
b
†
ˆ
b
+ ̄
hg
0
ˆ
a
†
1
ˆ
a
1
(
ˆ
b
+
ˆ
b
†
)
,
(2)
where
J
is the photon tunneling rate between the two
local microwave cavity modes, ˆ
a
j
(ˆ
a
†
j
) are the annihila-
tion (creation) photon operators of local microwave cav-
ity mode
j
, and
ˆ
b
(
ˆ
b
†
) is the annihilation (creation) op-
erator of the mechanical mode of the acoustic cavity.
Diagonalizing this Hamiltonian in a basis of even and
odd superpositions of the local microwave cavity modes,
and linearizing the interaction for the special case of
a strong red-sideband microwave pump field [10] yields
ˆ
H
′
em
= ̄
hG
(ˆ
a
+
ˆ
b
†
+ˆ
a
†
+
ˆ
b
), where
G
is the parametrically en-
hanced electromechanical coupling rate,
G
=
g
0
,
±
√
n
d,
−
,
g
0
,
±
=
g
0
/
2, and
n
d,
−
is the intra-cavity photon num-
ber due to the strong pump. In this scenario efficient
microwave pumping is realized for 2
J
≈
ω
m
, correspond-
ing to two-photon resonance, and the microwave pump
is at a drive frequency of
ω
d
=
ω
r,
+
−
ω
m
≈
ω
r,
−
,
where
ω
r,
+
(
ω
r,
−
) is the frequency of the upper even-
symmetry (lower odd-symmetry) supermode of the mi-
crowave cavity. The resulting electromechanical back-
action scattering rate between microwave photons and
acoustic phonons is
γ
em
= 4
G
2
/κ
+
, where
κ
+
is the de-
cay rate of the even-symmetry cavity mode (see App. C
for full derivation).
Figure 3(a) shows a scanning electron microscope
(SEM) image of a fabricated version of the double-cavity
device. This device is fabricated using an Al-on-SOI
process introduced in Ref. [36], and consists of two LC
lumped element microwave resonators which are induc-
tively coupled to each other and capacitively coupled to
a pair of hypersonic phononic crystal cavities of slightly
different (5 MHz) design frequency. The Al layer is de-
posited using electron-beam evaporation, and patterning
of the Si membrane and Al wiring is performed using
electron beam lithography and a combination of plasma
dry etching and lift-off. A SOI wafer with high resistiv-
ity (
>
∼
5 kΩ) Si device and handle layers is used to re-
duce the microwave losses, and the buried-oxide (BOX)