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
(ˆ
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
(ˆ
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
(ˆ
a
−
ˆ
a
†
)
2
+
V
2
DC
+ 2
iV
zpf
V
DC
(ˆ
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
(ˆ
a
ˆ
b
†
−
ˆ
a
†
ˆ
b
) =
i
ℏ
g
em
(ˆ
a
ˆ
b
†
−
ˆ
a
†
ˆ
b
)
.
(4)
This Hamiltonian has the form of an artificial piezoelectric response with an interaction strength
ℏ
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/
ℏ
=
ω
r
ˆ
a
†
ˆ
a
+
ω
m
ˆ
b
†
ˆ
b
+
ig
em
(ˆ
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
=
−
(
iω
r
+
κ/
2) ˆ
a
−
g
em
ˆ
b
−
√
κ
e
ˆ
a
in
(7)
̇
ˆ
b
=
−
(
iω
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)
(
iδ
+
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
iδ
+
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
iδ
+
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
ℏ
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
ℏ
/
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
/ε
1
−
1
/ε
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.
ℏ
g
em
≈
x
zpf
I
dA