Supplemental Document
Electro-optic transduction in silicon via
gigahertz-frequency nanomechanics:
supplement
H
AN
Z
HAO
,
1,2
A
LKIM
B
OZKURT
,
1,2
AND
M
OHAMMAD
M
IRHOSSEINI
1,2,
∗
1
The Gordon and Betty Moore Laboratory of Engineering, California Institute of Technology, Pasadena,
California 91125, USA
2
Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California
91125, USA
∗
mohmir@caltech.com
This supplement published with Optica Publishing Group on 20 June 2023 by The Authors under
the terms of the Creative Commons Attribution 4.0 License in the format provided by the authors
and unedited. Further distribution of this work must maintain attribution to the author(s) and the
published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.23226758
Parent Article DOI: https://doi.org/10.1364/OPTICA.479162
Supplementary Information: Electro-optic transduction in silicon via GHz-frequency
nanomechanics
Han Zhao,
1, 2
Alkim Bozkurt,
1, 2
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
(Dated: May 25, 2023)
I. DERIVATION OF THE ELECTROMECHANICAL CONVERSION
We can express the current passing through a motion-dependent capacitor,
C
(
x
), as [1]
i
(
t
) =
C
(
x
)
d
V
d
t
+
d
C
d
x
d
x
d
t
V
=
i
conductive
+
i
motional
.
(1)
As evident, the product of velocity and voltage gives rise to the motional current, which in the most general case
includes multiple frequency components. Assuming a static voltage bias of
V
b
, we find the RF component of the
motional current as
i
motional
,
RF
(
t
) =
V
b
d
C
d
x
d
x
d
t
.
(2)
The rate of energy loss of the mechanical resonator from the motional current can be written as
P
em
=
Z
0
V
b
2
d
C
d
x
2
d
x
d
t
2
,
(3)
where
Z
0
is the impedance of the microwave waveguide. This energy loss rate can be readily converted to an
electromechanical dissipation rate upon division by the total energy,
E
m
, stored in the mechanical oscillator
γ
em
=
P
em
E
m
=
Z
0
V
b
2
m
eff
d
C
d
x
2
.
(4)
Here, we have used
E
m
=
m
eff