of 16
nature physics
https://doi.org/10.1038/s41567-023-02080-w
Artic�e
A quantum electromechanical interface for
long-lived phonons
In the format provided by the
authors and unedited
Supplementary Information: A quantum electromechanical interface for long-lived
phonons
Alkim Bozkurt,
1, 2
Han Zhao,
1, 2
Chaitali Joshi,
1, 2
Henry G. Leduc,
3
Peter K. Day,
3
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
3
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109
(Dated: April 17, 2023)
CONTENTS
I. Electrostatic Interaction
1
A. Hamiltonian Derivation
1
B. Coherent Response
2
C. Coupling Perturbation Theory
2
D. Equivalent Circuit Model
3
II. Microwave Devices
4
A. Microwave Resonator
4
B. Purcell Filter
5
III. Mechanics Design
6
IV. Thermometry
7
A. Theory
7
B. Driven Response
9
C. Line Calibration
10
D. Pull-in Instability
10
V. Mechanical Coherence
11
A. Ringdown Measurements
11
B. Linewidth Characterization
12
C. TLS Model
13
References
14
I. ELECTROSTATIC INTERACTION
A. Hamiltonian Derivation
The interaction term in the Hamiltonian for a capacitor with mechanically moving electrodes (
C
m
) is given as
H
int
=
ˆ
q
2
2
C
m

1
C
m
∂C
m
∂x
ˆ
x

.
(1)
The displacement operator can be written as ˆ
x
=
x
zpf
(
ˆ
b
+
ˆ
b
). We have both electrostatic charges due to the
external voltage source and RF charge associated with the microwave resonance on top of the capacitor. This leads to
a charge operator which can be written as ˆ
q
=
iQ
zpf
a
ˆ
a
) +
Q
DC
. Inserting these operators into the Hamiltonian,
we obtain
H
int
=
1
2
∂C
m
∂x
x
zpf

i
Q
zpf
C
m
a
ˆ
a
) +
Q
DC
C
m

2
(
ˆ
b
+
ˆ
b
)
.
(2)
2
Noting that
Q
zpf
/C
m
=
V
zpf
,
Q
DC
/C
m
=
V
DC
and expanding the terms
H
int
=
1
2
∂C
m
∂x
x
zpf
(
ˆ
b
+
ˆ
b
)

V
2
zpf
a
ˆ
a
)
2
+
V
2
DC
+ 2
iV
zpf
V
DC
a
ˆ
a
)

.
(3)
Keeping only the interaction terms between the DC voltage and the RF fields and carrying out the rotating wave
approximation for our case of mechanics in resonance with microwave gives us
H
int
=
i
∂C
m
∂x
x
zpf
V
zpf
V
DC
a
ˆ
b
ˆ
a
ˆ
b
) =
i
g
em
a
ˆ
b
ˆ
a
ˆ
b
)
.
(4)
This Hamiltonian has the form of an artificial piezoelectric response with an interaction strength
g
em
=
∂C
m
∂x
x
zpf
V
zpf
V
DC
.
(5)
This constitutes a parametric interaction where the interaction strength scales linearly with the applied external
voltage.
Upon obtaining the interaction term, we can write the full Hamiltonian of our system as
H/
=
ω
r
ˆ
a
ˆ
a
+
ω
m
ˆ
b
ˆ
b
+
ig
em
a
ˆ
b
ˆ
a
ˆ
b
)
.
(6)
B. Coherent Response
In probing our cavity-mechanics system, we use a microwave tone. We can write the Langevin equations for eq. (6)
as
̇
ˆ
a
=
(
r
+
κ/
2) ˆ
a
g
em
ˆ
b
κ
e
ˆ
a
in
(7)
̇
ˆ
b
=
(
m
+
γ/
2)
ˆ
b
+
g
em
ˆ
a.
(8)
where we have neglected the thermal fluctuations entering the microwave and mechanics, as we are focused on the
coherent response of our system.
Taking the Fourier transform of the Langevin equations, we obtain
(
i
∆ +
κ/
2) ˆ
a
(
ω
) =
g
em
ˆ
b
(
ω
)
κ
e
ˆ
a
in
(
ω
)
(9)
(
+
γ/
2)
ˆ
b
(
ω
) =
g
em
ˆ
a
(
ω
)
,
(10)
where ∆ =
ω
r
ω
and
δ
=
ω
m
ω
. Substituting eq. (10) into eq. (9), we can express the microwave cavity operator
as
ˆ
a
(
ω
) =
κ
e
ˆ
a
in
(
ω
)
i
∆ +
κ/
2 +
g
2
em
+
γ/
2
.
(11)
Using input-output theory, we can define the output operator as ˆ
a
out
(
ω
) = ˆ
a
in
(
ω
) +
κ
e
ˆ
a
(
ω
). Using eq. (11), we
obtain an expression similar to the familiar electromechanically induced transparency (EIT) expression
ˆ
a
out
(
ω
)
ˆ
a
in
(
ω
)
= 1
κ
e
i
∆ +
κ/
2 +
g
2
em
+
γ/
2
.
(12)
We utilize this expression to fit the reflection traces we obtain of the cavity-mechanics system.
C. Coupling Perturbation Theory
Within the framework of cavity electromechanics, we can consider our interaction to be caused by radiation pressure.
More precisely, the stored electrical energy in our system changes with the mechanical displacement via the modulation
of the capacitance. Looking at the term in the Hamiltonian that leads to electrostatic interaction in eq. (3), we can
see that the change in the cross-electrical energy is the origin of this interaction and thus can be used to capture the
3
change in the capacitance. It is possible to express this change in the energy via a perturbative integral, similar to
the moving boundary integrals for electromechanical systems [1]. In this perturbative approach, we assume that the
displacement of the material boundaries does not change the electric field, but alters the local permittivity leading to
an electromechanical coupling rate
g
em
=
x
zpf
I
dA

Q
(
r
)
·
ˆ
n
)(∆
ε
E
(
r
)
DC
E
(
r
)
RF
ε
1
D
(
r
)
DC
D
(
r
)
RF

.
(13)
Here,
x
zpf
=
p
/
2
m
eff
ω
m
is the zero point fluctuations of displacement and
m
eff
is the effective mass of the acoustic
resonator.
Q
(
r
) is the normalized displacement where max[
Q
(
r
)] = 1. ∆
ε
=
ε
1
ε
2
and ∆
ε
1
= 1
1
1
2
are the
electrical permittivity contrast between the two materials that are on the boundary covered by the surface integral.
E
(
r
)
DC
(
E
(
r
)
RF
) is the parallel electric field component obtained from electrostatic simulations of the capacitor
with
V
DC
(
V
zpf
) applied to the capacitors. Likewise,
D
(
r
)
DC
(
D
(
r
)
RF
) is the perpendicular displacement field
obtained from the same simulation. In this expression, the voltage dependence of the coupling is directly embedded in
the capacitor voltages used in the simulations. One can alternatively solve for a given voltage (such as 1 V) and then
scale the fields appropriately based on
V
DC
and
V
zpf
. For instance,
g
0
can be simply obtained by setting
V
DC
= 1 V.
It has been previously observed that the electrical response of the capacitor at DC and RF frequencies may differ
from one another [2]. Generally, the charge carriers freeze off at cryogenic temperatures, giving rise to massive
resistivity values for silicon [3]. However, in capacitor structures under DC voltages, band bending leads to the
formation of a narrow space charge region which effectively screens out the field at the bulk of the silicon [4]. This
leads to a DC response which can approximately be modeled by treating silicon as a perfect conductor. On the other
hand, our microwave field which oscillates at a frequency above the RC cutoff cannot be screened and silicon behaves
like a perfect insulator for these fields.
Taking this into account, only fields perpendicular to the boundaries will exist for the DC field and the integral for
the electromechanical coupling can be simplified to.
g
em
x
zpf
I
dA
Q
(
r
)
·
ˆ
n
)(
ε
1
0
D
(
r
)
DC
D
(
r
)
RF

.
(14)
For our devices with
g
0
/
2
π
= 45
.
4 kHz/V this approach gives us an accurate simulation result of 46 kHz/V. If we
were to assume that silicon was an insulator at all frequencies and the field distributions were identical, we would
underestimate our coupling strength and obtain 32 kHz/V.
D. Equivalent Circuit Model
C
m
C
k
C
r
L
r
L
k
Z
0
V
DC
FIG. 1.
Equivalent circuit.
Equivalent electrical circuit of our system.
The calculation of the electromechanical coupling strength via finite element method simulations is sufficient to
completely describe the behavior of our system which consists of two coupled resonators. However, obtaining an
electrical equivalent circuit for the mechanical resonator is crucial for making the analysis of complex electromechanical
circuits more tractable. For this purpose, we model our mechanical resonator by a parallel LC resonator (
C
k
,
L
k
) in
series with a capacitor
C
m
. This model can be seen in fig. 1, where the coupling to the lumped element microwave
resonator (
C
r
,
L
r
) is capacitive via
C
m
.
4
The equivalent mechanical capacitance can be expressed as
C
k
=
C
2
m
ω
m
2 (
x
C
m
x
zpf
V
DC
)
2
C
m
.
(15)
The equivalent mechanical inductance can be obtained following the calculation of
C
k
by noting that
ω
m
= [
L
k
(
C
k
+
C
m
)]
1
/
2
. This circuit model gives us the correct expression for the electromechanical coupling strength, which is
attained by capacitive coupling between the two circuit modes where
g
em
=
C
m
2
p
(
C
k
+
C
m
)(
C
r
+
C
m
)
ω
r
ω
m
.
(16)
In this capacitive coupling picture, the value of
C
r
primarily sets the zero point fluctuations of voltage for the
microwave resonator since we have a small electromechanical participation ratio (
η
=
C
m
/
(
C
m
+
C
r
)
1). The small
participation ratio of our capacitor is caused by the small physical dimensions of our electromechanical capacitor
which is commensurate with 1
μ
m transverse acoustic wavelength at 5 GHz. This leads to the electrical energy on top
of the electromechanical capacitor being substantially diluted compared to the total electrical energy stored on the
microwave resonator.
We express the circuit parameters for device B in Table I. We note that the equivalent circuit model parameters for
the mechanical resonator are dependent on the external voltage bias
V
DC
. This scaling is noted in the table, where
L
k
and
C
k
are provided for 1 V.
TABLE I. Equivalent circuit parameters for device B. The mechanical parameters
C
k
and
L
k
are dependent on the applied
external voltage. The given values are for 1 V and the dependence on
V
DC
is specified.
Parameter
Value
C
m
0.2 fF
C
r
12.1 fF
L
r
75.8 nH
η
1.5 %
L
k
43.5 fH (1V
/V
DC
)
2
C
k
21.1 nF (
V
DC
/
1V)
2
f
r,m
5.26 GHz
g
0
/
2
π
45.4 kHz/V
In the circuit model, one can see that the mechanical resonator is capacitively coupled to the CPW which has an
impedance
Z
0
= 50 Ω. This leads to some direct external decay to the CPW. However, even at the maximum voltage
we can apply of 25 V, due to our massive equivalent capacitance this readout is at the level of 5 Hz, which is negligible
and further emphasizes the need for a microwave cavity to enhance electromechanical readout of mechanics.
II. MICROWAVE DEVICES
A. Microwave Resonator
The
λ/
4 tunable microwave resonators in our work are formed by ICP-RIE etching of TiN films (
t
15 nm) with
a sheet kinetic inductance of 40 pH
/
[5], which are sputtered on high resistivity (
>
3 kΩ
.
cm) SOI substrates (
device layer thickness 220 nm, manufactured by SEH). The high kinetic inductance TiN films are chosen for two
main reasons: (i) obtaining a large impedance (
Z
=
p
L/C
) resonator in order to enhance the electromechanical
interaction (
g
em
Z
) and (ii) attaining a high degree of tunability via an external magnetic field to bring the
microwave into resonance with mechanics. These goals are satisfied by forming a
λ/
4 resonator, with the inductive
component realized by a nanowire. The total kinetic inductance in this structure is a function of film properties and
geometry
L
k
=
L
8
π
2
l
w
,
(17)
where
L
is the sheet inductance,
l
is the nanowire length and
w
is the nanowire width. In order to maximize
the impedance, we etch our nanowires to be as narrow as possible, which in our case corresponds to a width of
5
0
2
4
6
8
10
B (mT)
-300
-200
-100
0
f
(MHz)
FIG. 2.
Microwave tuning.
Change in the microwave resonance frequency based on the applied external perpendicular
magnetic field. The solid line is a parabolic theory fit.
approximately 110 nm. Attempts to reduce wire width below this number led to reduced repeatability in fabrication
and a large disorder in resonator frequency.
A current passing through a TiN nanowire modifies the kinetic inductance in a nonlinear fashion
L
k
(
I
) =
L
k
(0)
"
1 +

I
I

2
#
.
(18)
where
I
is the critical current of the nanowire. Patterning of the nanowires to form closed loops and application
of an external perpendicular magnetic field, provides a ‘wireless’ means of modifying the kinetic inductance via the
screening current induced through the loops [6]. This is the mechanism we us to tune the resonator’s frequency.
With these principles in mind, we utilize a ladder-like topology for our microwave resonator. Finite-element method
(FEM) simulations indicate that our kinetic inductance excellently matches eq. (17) and we have negligible geometric
inductance, leading to a kinetic inductance participation ratio of approximately unity (
>
98%). Furthermore, we
calculate the impedance of our resonator to be 2.5 kΩ. The calculated lumped element equivalent circuit parameters
for the microwave resonator of device B can be seen in Table I. Despite having a substantial impedance, due to
our mechanical capacitance being very small, we have a low electrical energy participation ratio for the mechanical
capacitor (
η
=
C
m
/
(
C
m
+
C
r
)) of 1.5%. This small participation ratio permits us to connect multiple electromechanical
capacitors to our microwave resonator.
Following fabrication, we find the resonance frequencies in good agreement with the device modeling, with a
random offset with an approximate mean value of 300 MHz, which we attribute to fabrication disorder in the narrow
nanowires. We test the tunability of our devices by applying external magnetic fields via currents passing through
a superconducting coil mounted on top of the sample box. For small fractional shifts (normalized to the original
frequency), we expect the frequency shift to be given by
f
=
kB
2
,
(19)
where
k
is a device dependent proportionality constant and
B
is the external perpendicular magnetic field amplitude
[6]. This parabolic dependence is verified for device B on fig. 2. We can tune our device by about 6% of their resonance
frequency without causing any substantial degradation of the microwave intrinsic quality factor beyond
κ
i
/
2
π
500
kHz.
B. Purcell Filter
In our design, we prioritize redundancy in terms of the number of mechanical resonators by attaching four electrome-
chanical capacitors to a single microwave resonator. This is done to mitigate the impact of frequency shifts caused
by fabrication disorder of the microwave resonators, where the different mechanical resonators span roughly 500 MHz
(the disorder in the mechanical frequencies is much smaller than the microwave disorder). However, this approach
leads to an increased capacitance between our microwave resonator and the CPW, which gives rise to undesirably
large external microwave decay. In order to increase the cooperativity of our system and reach the strong-coupling
regime, this decay channel has to be suppressed.
6
FSR1
FSR3
FSR2
L1
L2
L3
Switch
DUT
2x1
Combiner
Input/Output
FIG. 3.
Purcell filter.
Circuit diagram of the Purcell filter configuration. The input/output line is attached to a circulator
(not shown in the figure) used for investigation of the devices in reflection mode. The final port of the switch is not connected
to a transmission line. The transmission lines have different lengths (L1,L2,L3). Their response is depicted as Lorentzian
resonances separated by a given FSR that differs for each cable. DUT: Device under test.
b
X
Γ
Normalized k
x
0
4
8
Frequency (GHz)
5
7
9
11
13
15
Number of Cells
30
40
50
60
g
0
/2
(kHz/V)
50
60
70
80
90
Vacuum Gap (nm)
20
40
60
80
100
g
0
/2
(kHz/V)
a
c
d
x
y
z
FIG. 4.
Moving capacitor design.
(a) Unit cell of the inner electrode of the electrostatic transducer. The beam is oriented
along the x-direction. (b) Unit cell acoustic band diagram. (c) Disorder simulations of coupling strength vs number of unit
cells of the inner electrode for
σ
= 2 nm disorder and 55 realizations for each data point. The simulations are for a 70-nm
vacuum gap. The data is presented as the mean with 2 standard deviation errobars. (d) Average coupling strength vs vacuum
gap size for 9 unit cells. The simulated average values are in the presence of fabrication disorder. The fit is
gap
1
.
66
.
We utilize an off-chip Purcell filter for this purpose as depicted schematically in fig. 3. We place the filter in parallel
to our device, where the notch filter reduces the external coupling strength of the microwave resonator. For this
purpose, we make use of home-built microwave filters, realized as open transmission lines with a multi-pole spectrum.
To get approximate frequency matching between our devices and the filter, we use a mechanical switch that enables
us to choose among multiple filters with different free spectral ranges (FSR) between 200-500 MHz. In cases where
we get perfect frequency matching between a device and a filter resonance, we can reduce the external decay rate
κ
e
/
2
π
to sub-200 kHz levels. Furthermore, we have a port in our switch which is not connected to a transmission
line, permitting us to effectively turn our filter off.
III. MECHANICS DESIGN
The mechanical behavior of our structures is investigated via FEM simulations on COMSOL. The inner electrode
of the electrostatic transducer is patterned to form a phononic crystal cavity consisting of multiple identical unit cells,
with the unit cell depicted in fig. 4a. These unit cells are completely metallized with a 15-nm layer of TiN on top of a