of 13
Reviewers' comments:
Reviewer #1 (Remarks to the Author):
This paper provides a clear description of the design, fabrication and measurement of two-
dimensional optimechanical crystals that are cooled close to their mechanical quantum ground
state. Measurements are reported for the optical and mechanical loss, mechanical heating due to
the optical probe signal, and estimates made for the quantum cooperativity for optimechanical
state transfer.
This paper reports an astounding mechanical quality factor in excess of one billion at 10 GHz,
reproducing, and closely parallel to, an earlier manuscript from the same group (ref 32), which
dealt with low temperature behavior in one-dimensional optimechanical crystals. This earlier work,
with even higher reported mechanical Q's at similar frequencies, apparently has not yet been
published, which is unfortunate, as this earlier paper represents a signature result with a detailed
analysis very similar to this work. The complicated thermal modeling and behavior reported here is
quite similar to that earlier work.
In the submitted paper here a similar level of mechanical quality factor is achieved at the lowest
optical excitation powers as in Ref 32, with this measurement similarly confounded by heating due
to the laser probe. However in this two-dimensional design the heating is reduced sufficiently that
more promising performance is predicted for somewhat more optimized devices. The paper is
mostly very clear and shows a number of new results; the supplementary also provides a wealth of
experimental detail that is quite welcome. I would support publication in Nature Communications
with some mandatory changes as below.
Please provide an explicit expression for the cooperativity C when it is introduced on p 1
Please provide an expression and reference for the extraction of the optimechanical coupling rate g
from the linewidth dependence on cavity photon number.
Many parameters are calculated from the data without any consideration for the uncertainties in
the arrived-at results; just as an example, the mechanical decay rates given on p 6 are given with
4 significant figures; I find it hard to believe, given the data, that these can be determined to 1
part in 10^4. Please provide uncertainties for all such derived quantities.
Reviewer #2 (Remarks to the Author):
The authors demonstrate a two-dimensional (2D) Si optomechanical crystal (OMC) cavity with a
high quantum cooperativity at millikelvin temperatures. The structure is well band-engineered in
both photonic and phononic systems and achieves high optical and mechanical quality factors. This
group investigated similar topic in a 1D nanobeam OMC structure before. In this study, design and
fabrication of a 2D OMC cavity with a high acoustic cavity Q and much better thermal conductance
with the cold bath reservoir result in a higher quantum cooperativity, which is the most relevant
figure-of-merit for quantum optomechanical applications. The progress from the 1D to 2D OMC
cavity is clear as stated in the last paragraph in p. 1, and this work achieves an effective quantum
cooperativity of unity. The level of experiments and analyses are high, and the discussion is
reasonable and convincing. Therefore, I recommend publication after minor revision.
1. Regarding the quantum cooperativity, the relevant threshold for coherent photon-phonon
interaction is unity. How high quantum cooperativity is necessary for realistic use in applications?
Please pick up one example of application and discuss how far/close current technology level is.
2. In Fig. 4b, np is proportional to nc^0.3. What determines this power-law of 0.3? Is it reasonable
that 1D and 2D have the same power-law?
3. (minor) In Fig. 1c, the broken lines are too thin and could not find them when I printed. I
recommend making it thicker.
Reviewer #3 (Remarks to the Author):
I have read the manuscript « Two-Dimensional Optomechanical Crystal Cavity with High Quantum
Cooperativity” from Prof Painter and co-workers. Briefly, the manuscript describes a new type of
two-dimensional optomechanical crystal cavity engineered for improving the low temperature
thermalization properties and thereby maximize the optomechanical cooperativity (which
represents the weight of quantum fluctuations respective to that of classical ones). This work
reports on the background, design and fabrication of the device, as well as on the optomechanical
and thermal experimental characterizations and calibration. The context and hypothesis of the
work are remarkably clear, as well as the adopted scientific methodology. The quality of the
experimental results and their agreement with theoretical modelling are high. This manuscript
represents an important piece of work to the field and beyond, thanks to the general problematic
being addressed (heat bath engineering applied to the design of quantum coherent devices) and
its very accessible presentation. In light of the above remarks, I strongly recommend publication
of this major piece of work in Nature Communications.
The manuscript is organized into 6 parts: The authors thoroughly introduce the context of their
work in the first part. Details on the design methodology and fabrication are given in the second
part, along with the presentation of the experimental setup. The optomechanical characterization
and calibration is provided in the third part. The fourth part presents the measurement and
analysis of the thermal properties of the optomechanical system under continuous optical driving.
The fifth part reports the measurement of the phonon occupation and corresponding effective
quantum cooperativity under continuous optical drive. The authors conclude in the last, discussion
part.
Introduction
In the first, introductory part, the authors set their work into the general context of (quantum)
optomechanical systems. They clearly describe the importance of developing high cooperativity
optomechanical systems operating at ultra-low temperatures, notably in the perspective of
building quantum hybrid interfaces between microwave frequency logic circuits and optical
quantum communication channels. The intrinsic problematic of 1-dimension optomechanical
crystal cavity systems, showing very high optomechanical coupling rates but reduced thermal
conductivity (and which therefore greatly suffer from the absorption-induced heating) is very well
introduced.
Comment: I have no specific comment on this very well written part.
Design/fabrication
In the second part, the authors present the concept of their new device aiming at preserving a
very high optomechanical coupling while simultaneously increasing the thermal conductivity,
enabling to load significantly more intra-cavity photons, and therefore resulting in a much
increased effective optomechanical cooperativity. Their strategy relies on making use of the
frequency-dependent density of phonon states within a 2D phononic bandgap structure consisting
of a 2D optomechanical crystal cavity. The fabrication steps, the analysis of the experimentally
measured geometrical properties and the simulated optomechanical parameters are clearly
presented in that section, confirming the relevance of the proposed approach as far as the
optomechanical properties are concerned.
Comment: I have a minor comment regarding Fig 2.c: I would advise the authors to mind the use
of colours (and to maybe compliment it with different point styles) for the colours-blind readership.
Optomechanical characterization
In this third part, the authors present the optomechanical characterization of the newly fabricated
devices. Optomechanical characterization is reported both at room and cryogenic temperature,
including the optical Q-factor, mechanical resonance frequency, optomechanical coupling rate and
mechanical Q-factor. In particular, the authors pay a great deal of attention for avoiding dynamical
backaction effects by performing ringdown measurements, which besides showing decreased
sensitivity towards dephasing, enables to be “in the dark”, thereby suppressing the contribution of
any delayed dynamical backaction. The authors notably report a massive mechanical Q-factor
exceeding 1 billion, with a mechanical resonance frequency in the 10 GHz range.
Comment: I have a comment on this part: The authors report measurements relying on non-linear
optomechanical amplification of a resonant phase modulation, which is sometimes also referred to
as “OMIT” measurement. I believe such experiment to be nontrivial to a broad readership and
would suggest a more pedagogic presentation, besides the (rightfully) given references. Along
these lines, I would recommend a description of the solid line fits featured on Fig. 3(b).
Thermal measurements
In this fourth part, the authors present a study of the thermal properties (both effective damping
rate and phonon occupation) of the OMC cavity as a function of the input optical power. The
authors essentially identify two regimes for the sensitivity of the mechanical properties as a
function of the intracavity photon number. Importantly, they establish a connection between the
effective temperature (number of phonons) and the mechanical damping rate, thereby describing
the effect of the absorption as that of an effective “hot bath”.
Comments: I have a few comments on this part.
a) I find the last sentence of the second paragraph (right column) on page 5 somehow too long
and not easy to understand. I would recommend the authors to try to simplify this sentence. On
the power low: could the authors maybe comment on a more fundamental solid-state physics
point of view maybe?
b) 3rd paragraph right column on page 5: the authors refer to Fig. 3(b) instead of Fig. 4(b).
c) page 6 right column: The authors state “at the lowest power (...) the linewidth saturates to a
constant value; this is not entirely clear to me that the data confirm this.
d) On \gamma_\phi: how do the authors prove this to be dephasing (besides being much larger
than the “zero power” ring down value)? Did the authors maybe perform ringdown measurements
(e.g. by means of a pump-probe configuration)? Could the authors better explain why the two
regimes of temperature are fitted using different models? Where do the authors set the “cut off”
between these two regimes?
e) End of 2nd paragraph right column page 6: The last sentence sounds somehow cryptic.
Measurement of the optomechanical cooperativity
In the fifth part, the authors report on optomechanical thermometry measurements, and
corresponding optomechanical cooperativity. Their notably use their analysis to attribute the
(significant) excess of thermal occupation to the phonon population within the coupling waveguide,
which contaminates that of the cavity under the effect of optical absorption. The (somehow
empiric) model of the authors shows very satisfying agreement, which certainly paves the way not
only to further technological improvement but for deeper understanding and assessment of the
spatial location of decoherence in ultra-sensitive optomechanical systems.
Comments: I have a comment/question on this part: Page 7 left column 3rd paragraph : did the
authors tried to optimize the sideband ratio to see if this could be beneficial to the cooperativity?
Discussion
In the last part, the authors briefly discuss their result and put them into perspective with future
possible research pathways that could benefit from them. This part is convincing and very well
written.
Comment: I have one minor comment: An 68 fold increase of the thermal conductivity is
mentioned, which disagrees with the number (42) given in the main text body.
Author Response:
The
authors
thank
all
the
reviewers
for
their
careful
and
detailed
review
of
our
manuscript.
Please
find
below
the
authors’
responses
(
marked
in
red
)
to
each
of
the
reviewers’ points (
in black
).
Reviewer #1 (Remarks to the Author):
This
paper
provides
a
clear
description
of
the
design,
fabrication
and
measurement
of
two-dimensional
optomechanical
crystals
that
are
cooled
close
to
their
mechanical
quantum
ground
state.
Measurements
are
reported
for
the
optical
and
mechanical
loss,
mechanical
heating
due
to
the
optical
probe
signal,
and
estimates
made
for
the
quantum cooperativity for optomechanical state transfer.
This
paper
reports
an
astounding
mechanical
quality
factor
in
excess
of
one
billion
at
10
GHz,
reproducing,
and
closely
parallel
to,
an
earlier
manuscript
from
the
same
group
(ref
32),
which
dealt
with
low
temperature
behavior
in
one-dimensional
optomechanical
crystals.
This
earlier
work,
with
even
higher
reported
mechanical
Q's
at
similar
frequencies,
apparently
has
not
yet
been
published,
which
is
unfortunate,
as
this
earlier
paper
represents
a
signature
result
with
a
detailed
analysis
very
similar
to
this
work.
The
complicated
thermal
modeling
and
behavior
reported
here
is
quite
similar
to
that
earlier work.
In
the
submitted
paper
here
a
similar
level
of
mechanical
quality
factor
is
achieved
at
the
lowest
optical
excitation
powers
as
in
Ref
32,
with
this
measurement
similarly
confounded
by
heating
due
to
the
laser
probe.
However
in
this
two-dimensional
design
the
heating
is
reduced
sufficiently
that
more
promising
performance
is
predicted
for
somewhat
more
optimized
devices.
The
paper
is
mostly
very
clear
and
shows
a
number
of
new
results;
the
supplementary
also
provides
a
wealth
of
experimental
detail
that
is
quite
welcome.
I
would
support
publication
in
Nature
Communications
with
some
mandatory changes as below.
Please provide an explicit expression for the cooperativity C when it is introduced on p 1
[Authors’
Response]:
Thanks
for
pointing
this
out.
We
have
added
an
expression
for
the
cooperativity C when it is introduced in the fourth paragraph on p1.
Please
provide
an
expression
and
reference
for
the
extraction
of
the
optimechanical
coupling rate g from the linewidth dependence on cavity photon number.
[Authors’
Response]:
Thanks
for
the
comment.
An
expression
and
reference
for
the
extraction
of
the
optomechanical
coupling
rate
g
from
the
linewidth
dependence
on
cavity
photon
number
(Fig.
3(b)
solid
line
fit)
have
been
added
in
the
manuscript
(first
paragraph of page 4).
Many
parameters
are
calculated
from
the
data
without
any
consideration
for
the
uncertainties
in
the
arrived-at
results;
just
as
an
example,
the
mechanical
decay
rates
given
on
p
6
are
given
with
4
significant
figures;
I
find
it
hard
to
believe,
given
the
data,
that
these
can
be
determined
to
1
part
in
10^4.
Please
provide
uncertainties
for
all
such
derived quantities.
[Authors’
Response]:
Yes,
we
agree
with
the
reviewer.
We
have
added
uncertainties
to
key
device
parameters
with
90%
confidence
interval.
For
example,
mechanical
decay
rate is reported as 8.28(+1.25,-0.43)~Hz now.
Reviewer #2 (Remarks to the Author):
The
authors
demonstrate
a
two-dimensional
(2D)
Si
optomechanical
crystal
(OMC)
cavity
with
a
high
quantum
cooperativity
at
millikelvin
temperatures.
The
structure
is
well
band-engineered
in
both
photonic
and
phononic
systems
and
achieves
high
optical
and
mechanical
quality
factors.
This
group
investigated
similar
topic
in
a
1D
nanobeam
OMC
structure
before.
In
this
study,
design
and
fabrication
of
a
2D
OMC
cavity
with
a
high
acoustic
cavity
Q
and
much
better
thermal
conductance
with
the
cold
bath
reservoir
result
in
a
higher
quantum
cooperativity,
which
is
the
most
relevant
figure-of-merit
for
quantum
optomechanical
applications.
The
progress
from
the
1D
to
2D
OMC
cavity
is
clear
as
stated
in
the
last
paragraph
in
p.
1,
and
this
work
achieves
an
effective
quantum
cooperativity
of
unity.
The
level
of
experiments
and
analyses
are
high,
and
the
discussion
is
reasonable
and
convincing.
Therefore,
I
recommend
publication after minor revision.
1.
Regarding
the
quantum
cooperativity,
the
relevant
threshold
for
coherent
photon-phonon
interaction
is
unity.
How
high
quantum
cooperativity
is
necessary
for
realistic
use
in
applications?
Please
pick
up
one
example
of
application
and
discuss
how far/close current technology level is.
[Authors’
Response]:
One
example
application
for
cavity-optomechanical
systems
is
phonon-mediated
quantum
state
transduction
between
microwave
and
optical
photons.
We
have
added
a
section
in
the
Supplementary
Information
discussing
the
bi-directional
transduction
efficiency
and
signal-to-noise
ratio.
This
example
highlights
the
fact
that
already
at
a
Ceff
>
1,
one
to
in
principle
can
realize
single
photon
conversion
with
an
SNR > 1.
2.
In
Fig.
4b,
np
is
proportional
to
nc^0.3.
What
determines
this
power-law
of
0.3?
Is
it
reasonable that 1D and 2D have the same power-law?
[Authors’
Response]:
This
is
a
good
point.
The
power
law
exponent
α
is
equal
to
the
effective
number
of
spatial
dimensions
d
of
the
material/structure
under
consideration.
Effectively,
the
hot
phonon
bath
radiates
energy
as
a
black
body,
with
radiated
power
scaling
as
T_p^(α+1)
via
Planck’s
law,
where
T_p
is
the
effective
temperature
of
the
“hot
bath”.
In
the
case
of
a
structure
with
2-dimensional
phonon
density
of
states,
α
=
d
=
2
and
the
hot
phonon
bath
occupancy/temperature
scales
as
n_p
~
T_p
~
Pin^(1/3)
~
n_c^(1/3).
This
approximate
scaling
is
expected
to
be
valid
so
long
as
phonons
in
the
hot
phonon
bath
approximately
thermalize
with
each
other
upon
creation
from
optical
absorption
events,
and
then
radiate
freely
(ballistically)
into
the
effective
zero
temperature substrate.
The
density
of
states
for
1D
and
2D
OMC
cavities
are
both
determined
by
the
nano-structure
of
the
devices
and
the
frequency
of
the
phonons
involved
in
heat
transport.
Due
to
the
geometric
aspect
ratio
of
the
thin-film
(220nm
membrane),
the
local
density
of
phonon
states
becomes
restricted
at
lower
frequency,
decreasing
the
rates
of
phonon-phonon
scattering
at
low
frequency
relative
to
those
of
a
bulk
crystal
with
a
3D
Debye
density
of
states.
A
phonon
bottleneck
occurs
as
the
density
of
states
passes
from
3D
(continuum)
to
2D
due
to
this
reduction
of
the
phonon-phonon
scattering.
The
phonons
of
the
bath
tend
to
pile
up
at
these
frequencies,
and
as
such
most
of
the
phonons
that
contribute
to
thermal
conduction
will
be
around
20GHz
corresponding
to
an
acoustic
wavelength
of
the
thickness
of
the
Si
device
layer
(200nm).
This
Si
device
layer
thickness
is
the
smallest
dimension
of
both
the
1D
and
2D
OMCs,
with
the
lateral
dimension
of
the
1D
OMC
still
much
larger
(x5)
than
this
wavelength.
As
a
result,
both
the
1D
and
2D
OMCs
both
have
a
hot
phonon
bath
with
approximately
2D
density
of
states.
Numerical
simulations
of
the
acoustc
modes
of
the
1D
nanobeam
OMC
confirm
that
above
the
OMC
bandgap
frequencies
the
density
of
phonon
states
is
approximately
that
of
a
2D
plate.
This
detailed
analysis
is
presented
in
another
of
our
papers
which
we
reference,
arXiv:1901.04129
(2019),
Appendix
D.1
and
Appendix H.
3.
(minor)
In
Fig.
1c,
the
broken
lines
are
too
thin
and
could
not
find
them
when
I
printed. I recommend making it thicker.
[Authors’
Response]:
Thanks
for
pointing
this
out.
We
have
improved
Fig.
1c
(and
also
Fig. 1b) with thicker lines and higher contrast colors.
Reviewer #3 (Remarks to the Author):
I
have
read
the
manuscript
«
Two-Dimensional
Optomechanical
Crystal
Cavity
with
High
Quantum
Cooperativity”
from
Prof
Painter
and
co-workers.
Briefly,
the
manuscript
describes
a
new
type
of
two-dimensional
optomechanical
crystal
cavity
engineered
for
improving
the
low
temperature
thermalization
properties
and
thereby
maximize
the
optomechanical
cooperativity
(which
represents
the
weight
of
quantum
fluctuations
respective
to
that
of
classical
ones).
This
work
reports
on
the
background,
design
and
fabrication
of
the
device,
as
well
as
on
the
optomechanical
and
thermal
experimental
characterizations
and
calibration.
The
context
and
hypothesis
of
the
work
are
remarkably
clear,
as
well
as
the
adopted
scientific
methodology.
The
quality
of
the
experimental
results
and
their
agreement
with
theoretical
modelling
are
high.
This
manuscript
represents
an
important
piece
of
work
to
the
field
and
beyond,
thanks
to
the
general problematic being addressed (heat bath engineering applied to the
design
of
quantum
coherent
devices)
and
its
very
accessible
presentation.
In
light
of
the
above
remarks,
I
strongly
recommend
publication
of
this
major
piece
of
work
in
Nature
Communications.
The
manuscript
is
organized
into
6
parts:
The
authors
thoroughly
introduce
the
context
of
their
work
in
the
first
part.
Details
on
the
design
methodology
and
fabrication
are
given
in
the
second
part,
along
with
the
presentation
of
the
experimental
setup.
The
optomechanical
characterization
and
calibration
is
provided
in
the
third
part.
The
fourth
part
presents
the
measurement
and
analysis
of
the
thermal
properties
of
the
optomechanical
system
under
continuous
optical
driving.
The
fifth
part
reports
the
measurement
of
the
phonon
occupation
and
corresponding
effective
quantum
cooperativity
under
continuous
optical
drive.
The
authors
conclude
in
the
last,
discussion part.
Introduction
In
the
first,
introductory
part,
the
authors
set
their
work
into
the
general
context
of
(quantum)
optomechanical
systems.
They
clearly
describe
the
importance
of
developing
high
cooperativity
optomechanical
systems
operating
at
ultra-low
temperatures,
notably
in
the
perspective
of
building
quantum
hybrid
interfaces
between
microwave
frequency
logic
circuits
and
optical
quantum
communication
channels.
The
intrinsic
problematic
of
1-dimension
optomechanical
crystal
cavity
systems,
showing
very
high
optomechanical
coupling
rates
but
reduced
thermal
conductivity
(and
which
therefore
greatly
suffer
from
the absorption-induced heating) is very well introduced.
Comment: I have no specific comment on this very well written part.
Design/fabrication
In
the
second
part,
the
authors
present
the
concept
of
their
new
device
aiming
at
preserving
a
very
high
optomechanical
coupling
while
simultaneously
increasing
the
thermal
conductivity,
enabling
to
load
significantly
more
intra-cavity
photons,
and
therefore
resulting
in
a
much
increased
effective
optomechanical
cooperativity.
Their
strategy
relies
on
making
use
of
the
frequency-dependent
density
of
phonon
states
within
a
2D
phononic
bandgap
structure
consisting
of
a
2D
optomechanical
crystal
cavity.
The
fabrication
steps,
the
analysis
of
the
experimentally
measured
geometrical
properties
and
the
simulated
optomechanical
parameters
are
clearly
presented
in
that
section,
confirming
the
relevance
of
the
proposed
approach
as
far
as
the
optomechanical properties are concerned.
Comment:
I
have
a
minor
comment
regarding
Fig
2.c:
I
would
advise
the
authors
to
mind
the
use
of
colours
(and
to
maybe
compliment
it
with
different
point
styles)
for
the
colours-blind readership.
[Authors’
Response]:
Thanks
for
pointing
this
out.
We
have
improved
Fig.
2c
by
tuning
the
colors
used
in
this
figure
to
a
more
color-blind-friendly,
as
well
as
using
different
symbols for each curve. We adjusted the description in the caption.
Optomechanical characterization
In
this
third
part,
the
authors
present
the
optomechanical
characterization
of
the
newly
fabricated
devices.
Optomechanical
characterization
is
reported
both
at
room
and
cryogenic
temperature,
including
the
optical
Q-factor,
mechanical
resonance
frequency,
optomechanical
coupling
rate
and
mechanical
Q-factor.
In
particular,
the
authors
pay
a
great
deal
of
attention
for
avoiding
dynamical
backaction
effects
by
performing
ringdown
measurements,
which
besides
showing
decreased
sensitivity
towards
dephasing,
enables
to
be
“in
the
dark”,
thereby
suppressing
the
contribution
of
any
delayed
dynamical
backaction.
The
authors
notably
report
a
massive
mechanical
Q-factor
exceeding 1 billion, with a mechanical resonance frequency in the 10 GHz range.
Comment:
I
have
a
comment
on
this
part:
The
authors
report
measurements
relying
on
non-linear
optomechanical
amplification
of
a
resonant
phase
modulation,
which
is
sometimes
also
referred
to
as
“OMIT”
measurement.
I
believe
such
experiment
to
be
nontrivial
to
a
broad
readership
and
would
suggest
a
more
pedagogic
presentation,
besides
the
(rightfully)
given
references.
Along
these
lines,
I
would
recommend
a
description of the solid line fits featured on Fig. 3(b).
[Authors’
Response]:
Thank
you
for
the
comment.
We
have
added
a
section
in
the
Supplementary
Information
explaining
the
concept
of
“OMIT”.
An
expression
and
description
for
the
extraction
of
the
optomechanical
coupling
rate
g
from
the
linewidth
dependence
on
cavity
photon
number
(Fig.
3(b)
solid
line
fits)
have
been
added
in
the
manuscript.
Thermal measurements
In
this
fourth
part,
the
authors
present
a
study
of
the
thermal
properties
(both
effective
damping
rate
and
phonon
occupation)
of
the
OMC
cavity
as
a
function
of
the
input
optical
power.
The
authors
essentially
identify
two
regimes
for
the
sensitivity
of
the
mechanical
properties
as
a
function
of
the
intracavity
photon
number.
Importantly,
they
establish
a
connection
between
the
effective
temperature
(number
of
phonons)
and
the
mechanical
damping
rate,
thereby
describing
the
effect
of
the
absorption
as
that
of
an
effective “hot bath”.
Comments: I have a few comments on this part.
a)
I
find
the
last
sentence
of
the
second
paragraph
(right
column)
on
page
5
somehow
too
long
and
not
easy
to
understand.
I
would
recommend
the
authors
to
try
to
simplify
this
sentence.
On
the
power
low:
could
the
authors
maybe
comment
on
a
more
fundamental solid-state physics point of view maybe?
[Authors’
Response]:
Point
taken.
We
have
worked
to
rephrase
and
simplify
this
sentence.
The
power
law
exponent
is
dependent
on
the
effective
number
of
spatial
dimensions
d
of
the
material/structure
under
consideration.
The
density
of
states
for
the
hot
phonon
bath
in
both
1D
and
2D
OMC
cavities
are
both
determined
by
the
smallest
dimension
of
each
structure,
thickness
of
the
Si
device
layer.
More
details
are
provided
in response to reviewer #1, comment 2.
b)
3rd
paragraph
right
column
on
page
5:
the
authors
refer
to
Fig.
3(b)
instead
of
Fig.
4(b).
[Authors’
Response]:
Thank
you
for
pointing
this
out.
This
was
an
oversight,
and
we
have corrected it from 3(b) to 4(b).
c)
page
6
right
column:
The
authors
state
“at
the
lowest
power
(...)
the
linewidth
saturates to a constant value; this is not entirely clear to me that the data confirm this.
[Authors’
Response]:
We
agree
with
the
reviewer,
the
wording
was
not
accurate.
The
numbers
measured
directly
do
not
get
into
the
lowest
power
regime
where
we
see
complete
saturation,
rather
the
dependence
on
n_c
begins
to
saturate.
We
have
made
this
more
explicit
in
the
revised
text
to
avoid
confusion.
However,
we
still
apply
in
our
model
gamma(n_c)
=
gamma_phi
+
gamma_p(n_c),
and
the
best
fit
of
this
model
gives
a residual linewidth of gamma_phi as indicated.
We
should
note
that
this
pure
dephasing
term
is
expected,
as
for
previous
experiments
with
a
similar
OMC
cavity
(arXiv:1901.04129
(2019)),
direct
measurements
of
mechanical
frequency
jittering
were
performed,
yielding
gamma_phi
of
a
similar
magnitude.
d)
On
\gamma_\phi:
how
do
the
authors
prove
this
to
be
dephasing
(besides
being
much
larger
than
the
“zero
power”
ring
down
value)?
Did
the
authors
maybe
perform
ringdown
measurements
(e.g.
by
means
of
a
pump-probe
configuration)?
Could
the
authors
better
explain
why
the
two
regimes
of
temperature
are
fitted
using
different
models? Where do the authors set the “cut off” between these two regimes?
[Authors’
Response]:
In
these
devices
we
did
not
prove
that
the
residual
linewidth
at
low
optical
pumping
was
indeed
due
to
dephasing
(frequency
jitter).
However,
as
noted
above,
for
very
similar
1D
OMC
devices
we
did
do
rapid
spectroscopic
measurements
using
a
pump-probe
technique,
and
we
were
able
to
measure
the
frequency
jitter
of
the
line directly (arXiv:1901.04129 (2019)).
Regarding
how
we
did
the
fitting
across
the
different
regimes,
we
have
added
a
section
in
the
Supplementary
Information
describing
these
details
and
explaining
how
the
cut-off is implemented in our fit.
e)
End
of
2nd
paragraph
right
column
page
6:
The
last
sentence
sounds
somehow
cryptic.