of 48
science.sciencemag.org/content/
370/
6518/
840
/suppl/DC1
Supplementary Materials for
Nano
-acoustic resonator with ultralong phonon lifetime
Gregory S. MacCabe*, Hengjiang Ren*, Jie Luo, Justin D. Cohen, Hengyun Zhou,
Alp Sipahigil, Mohammad Mirhosseini, Oskar Painter
*These authors contributed equally to this work.
†Corresponding author. Email: opainter@caltech.edu
Published
13 November
2020,
Science
370
, 840
(2020)
DOI:
10.1126/science.
abc7312
This PDF file includes:
Materials and Methods
Supplementary Text
Figs. S1 to S
19
Tables S1 to S
4
References
2
I. MATERIALS AND METHODS: DEVICE FABRICATION
The devices were fabricated using a silicon-on-insulator wafer with a silicon (Si) device layer thickness of 220 nm
and buried-oxide layer thickness of 3
μ
m. The device geometry was defined by electron-beam lithography followed by
inductively coupled plasma reactive ion etching (ICP-RIE) to transfer the pattern through the 220 nm Si device layer.
Photoresist was then used to define a ‘trench’ region of the chip to be etched and cleared for fiber access to device
waveguides. In the unprotected trench region of the chip, the buried-oxide layer is etched using a highly anisotropic
plasma etch, and the handle Si layer is cleared to a depth of 100
μ
m using an isotropic plasma etch. The devices were
then undercut using a vapor-HF etch and cleaned in a piranha solution before a final vapor-HF etch to remove the
chemically-grown oxide. In fabrication, devices were spatially grouped into arrays in which the number of acoustic
radiation shield periods is scaled from zero to ten while all other geometric parameters are held nominally identical.
3
II. MATERIALS AND METHODS: MODELING OF DISORDER IN THE OMC CAVITY
The use of a phononic bandgap shield is necessitated by the lack of a full gap for the nanobeam cavity. Finite-element
method (FEM) numerical modeling indicates that the addition of the cross shield provides significant protection against
nanometer-scale disorder which is inherently introduced during device fabrication. In Fig. S-1 we present a numerical
study of the effects of random fabrication imperfections on the radiative damping of the shielded OMC cavity mode.
We compare in Fig. S-1(b) the simulated acoustic
Q
-factor of the ideal, unperturbed cavity structure to that of cavity
structures with a fixed level of disorder in the hole sizes (standard deviation,
σ
size
= 4 nm) and varying levels of
disorder in the hole centers (
σ
pos.
= 2
,
4
,
8 nm). An absence of perturbations to the nanobeam cavity, even without
any shielding, yields large radiation-limited
Q
-factors in excess of 10
10
. This is a result of the quasi-bandgap that
exists in the nanobeam mirror section for modes of a specific symmetry about the center-line of the beam; however,
any perturbation that breaks this symmetry results in a compromised quasi-bandgap in the nanobeam (
Q
drops
from
&
10
10
to
.
10
5
for nanometer-scale perturbations). Conversely, the exponential trend of the radiation-limited
Q
-factor with the number of shield periods is consistently a factor of
×
5
.
5 per additional period, independent of the
disorder level. This interpretation is further bolstered by the plots in Fig. S-1(c) comparing the linear acoustic energy
density along the axis of the nanobeam,
W
, for the mode of the unperturbed cavity and the modes of two different
realizations of disordered cavities.
4
III. MATERIALS AND METHODS: MEASUREMENT SETUP, OPTICAL CHARACTERIZATION,
AND OPTOMECHANICAL CALIBRATION
A. Measurement Setup
The full measurement setup used for device characterization is shown in Fig. S-2. The light source is a fiber-coupled
tunable external-cavity diode laser, of which a small portion is sent to a wavemeter (
λ
-meter) for frequency stabi-
lization. The light is then sent to high-finesse tunable fiber Fabry-Perot filter (Micron Optics FFP-TF2, bandwidth
50 MHz, FSR 20 GHz) to reject laser phase noise at the mechanical frequency, which can contribute to noise-photon
counts on the SPDs. After this prefiltering, the light is routed to an electro-optic phase modulator (
φ
-mod) which
is driven by an RF signal generator at the mechanical frequency to generate optical sidebands used for locking
the detection-path filters. The light is then directed via 2
×
2 mechanical optical switches into a ”high-extinction”
path consisting of a series of modulator components which are driven by a digital pulse generator to generate high-
extinction-ratio optical pulses. The digital pulse generator is used to synchronize the switching of the modulation
components as well as to trigger the time-correlated single-photon-counting (TCSPC) module. Of these modulation
components, two are electro-optic intensity modulators which together provide
60 dB of fast extinction (
20 ns rise
and fall times), and two are Agiltron NS 1
×
1 switches (rise time 100 ns, fall time
30
μ
s) which provide a total of
36 dB of additional extinction. The total optical extinction used to generate our optical pulses is approximately 96
dB, which is greater than the cross-talk specification of our mechanical optical switches. For this reason we use two
2
×
2 switches in parallel to isolate the high-extinction path to ensure that our off-state optical power is limited by
our high-extinction modulation components rather than by cross-talk through the mechanical switches. The light is
then passed through a variable optical attenuator (VOA) to control the input pulse on-state power level to the cavity,
and sent to a circulator which directs the light to a lensed-fiber tip for end-fire coupling to devices inside a dilution
refrigerator. The reflected signal is then routed back to either one of two detection setups. The first includes an
erbium-doped fiber amplifier (EDFA) and a high-speed photodetector (PD) connected to a spectrum analyzer (SA)
and a vector network analyzer (VNA). The second detection path is used for the phonon counting measurements.
Here the light passes through three cascaded high-finesse tunable fiber Fabry-Perot filters (Micron Optics FFP-TF2)
inside an insulating housing and then to the SPD inside the dilution refrigerator.
The cascaded fiber Fabry-Perot (FP) filters are aligned to the optical cavity resonance frequency
ω
c
during mea-
surement such that the signal reaching the SPDs consists of sideband-scattered photons and a small contribution
of laser-frequency pump-bleed-through. In total the filters suppress the pump by
>
100 dB. This bleed-through is
calibrated by positioning the laser far off-resonance of the optical cavity, such that the device acts simply as a mirror,
while fixing the relative detuning of the filters and the pump laser at the mechanical frequency
ω
m
/
2
π
and measuring
the photon count rate on the SPDs as a function of laser power.
Additionally, both the FP-filters and the EOMs will drift during measurement and must be periodically re-locked.
We therefore regularly stop the measurement and perform a re-locking routine. First, we re-lock the EOMs by
applying a sinusoidal dithering signal of
1 V to them while monitoring the optical transmission, then decrease the
dithering amplitude gradually to lock to the minimum of transmission. Next we switch out of the high-extinction
pulse path (SW-2A,2B) and out of the SPD path (SW-5), drive the phase modulator with a large RF power at
ω
m
/
2
π
to generate large optical sidebands at the cavity resonance frequency, and send this light into the FP-filter stack. The
transmission through each filter is monitored while a dithering sinusoidal voltage is applied to each filter successively,
and the amplitude and DC offset of the dithering signal are adjusted until the optical transmission signal at the
desired sideband is maximized. The offset voltage is then held fixed during the subsequent measurement run. The
filters will drift due to both thermal fluctuations and acoustic disturbances in their environment, so in order to further
improve the filters’ stability we have placed them inside a custom-built insulated housing.
The SPDs used in this work are amorphous WSi-based superconducting nanowire single-photon detectors developed
in collaboration between the Jet Propulsion Laboratory and NIST. The SPDs are mounted on the still stage of
the dilution refrigerator at
700 mK. Single-mode optical fibers are passed into the refrigerator through vacuum
feedthroughs and coupled to the SPDs via a fiber sleeve attached to each SPD mount. The radio-frequency output of
each SPD is amplified by a cold-amplifier mounted on the 50 K stage of the refrigerator as well as a room-temperature
amplifier, then read out by a triggered PicoQuant PicoHarp 300 time-correlated single photon counting module. We
have observed SPD dark count rates as low as
0
.
6 c.p.s. and SPD quantum efficiency
η
SPD
'
60%.
B. Optical Characterization
Each device we have measured in this work was characterized optically in order to determine its optical resonance
frequency
ω
c
, total optical linewidth
κ
, waveguide-cavity coupling rate
κ
e
, waveguide-cavity coupling efficiency
η
κ
=
5
κ
e
, and fiber-to waveguide coupling efficiency
η
cpl
. In particular, the waveguide-cavity coupling efficiency
η
κ
is
measured by placing the laser far off-resonance and using the VNA to drive an intensity modulator to sweep an optical
sideband through the cavity frequency and measure the optical response on a high-speed PD (after amplification by
the EDFA) connected to the VNA signal port. From this we obtain the amplitude and phase response of the cavity,
which are fitted to determined
η
κ
.
C. Vacuum Optomechanical Coupling Rate and Mode Occupancy Calibration
The measurements presented in this work rely on calibration of the vacuum optomechanical coupling rate
g
0
. After
fitting the total optical linewidth
κ
from an optical reflection spectrum and calibrating
η
κ
as described above, the
intracavity photon number
n
c
for a specified detuning is known and is proportional to the optical power input to the
cavity:
n
c
=
ˆ
a
ˆ
a
=
P
in
~
ω
l
κ
e
2
+ (
κ/
2)
2
,
(S-1)
where
ω
l
is the applied laser frequency. In these phonon-counting measurements, a critical calibration parameter is
the photon scattering rate per phonon in the mechanical mode. In particular, from Ref. [16, 32, 33] the photon count
rate at the SPD for a red- or blue-detuned pump is:
Γ(∆ =
±
ω
m
) = Γ
dark
+ Γ
pump
+ Γ
SB,0
(
n
m
+
1
2
(1
1))
,
(S-2)
where
n
m
is the average phonon occupancy of the breathing mode, Γ
SB,0
=
η
det
η
cpl
η
κ
γ
OM
is the
detected
photon
scattering rate per phonon (including experimental set-up efficiencies) and
γ
OM
= 4
g
2
0
n
c
is the optomechanical
damping rate. In the absence of mechanical occupancy, pump photons may be spontaneously scattered by the
mechanics owing to the nonzero mechanical susceptibility at the pump frequency. These real photons scattered by
the mechanical vacuum noise in the presence of a pump laser are detected at a rate Γ
SB,0
, providing a calibration of
the per-phonon count rate directly to the vacuum noise. Here
η
det
is the measured overall detection efficiency of the
setup, including losses in the fiber runs and circulator, insertion losses in the filters, the fiber run inside the dilution
refrigerator, and the detection efficiency of the SPD. To calibrate Γ
SB,0
(for a fixed intracavity photon number
n
c
),
a blue-detuned pump (∆ =
ω
m
) pulsed with a repetition time of
T
per
drives the mechanics. In the initial time
bin during the pulse,
n
m

1 (if 1
/T
per

γ
0
) and we can approximate the sideband photon count rate Γ
Γ
SB,0
.
Including detection non-idealities such as bleed-through of the pump laser to the SPD and dark counts on the SPD,
the detected count rate is Γ(
T
pulse
= 0) = Γ
DCR
+ Γ
pump
+ Γ
SB,0
. This measurement provides an absolute calibration
of the detection photon count rate to the mechanical vacuum noise, where the count rate is proportional to intracavity
photon number, allowing calibrated thermometry with a precision that is independent of knowledge of the losses in
the optical path. Additional knowledge of the optical path losses enables the inference of
γ
OM
from a measurement
of Γ
SB,0
, which can be used to calibrate the vacuum optomechanical coupling rate. For one representative device as
shown in Figure S-3 we measure Γ
SB,0
= 3
.
263
×
10
3
c.p.s. using a measurement photon number of
n
c
= 101, and
with
κ
known from independent measurements we extract
g
0
/
2
π
= 713 kHz, consistent with previous measurements
on similar devices with similar orientations relative to the Si crystal axes [16].
An important figure of merit for phonon-counting measurements is the sensitivity, expressed as a noise-equivalent
phonon number
n
NEP
. This
n
NEP
represents the equivalent occupancy inferred from noise counts only:
n
NEP
=
Γ
dark
+ Γ
pump
Γ
SB,0
.
(S-3)
The acoustic breathing mode thermalizes to a mode temperature
T
m
which is related to the applied fridge tem-
perature
T
f
through the thermal conductance
G
th
of the structure. In order to measure the minimum temperature
T
b,min
to which the mode thermalizes at the lowest fridge temperature
T
f
= 10 mK, we use a low-power (
n
c
= 10)
red-detuned pulsed probe and a device having relatively low
Q
m
= 3
.
57
×
10
5
(chosen so that the mode thermalizes to
its base temperature rapidly between each incident optical pulse). The initial mode occupancy during the pulse then
approximately corresponds to the ‘off-state’ occupancy
n
0
. However, as the optical probe turns on during the first
several time bins of the pulse, the mode is heated such that the initial observed occupancy exceeds
n
0
. We therefore
extract
n
0
from a fit to the pulsed heating model which is extrapolated back to
t
= 0.
6
Figure S-4 shows a fit to the pulse occupancy in the on-state, which yields
T
m
35
.
6 mK. The lower bound of
10 mK is set by the minimum applied fridge temperature, and the upper bound 60 mK corresponds to the directly
observed occupancy value in the initial measurement bin.
D. Summary of Device Parameters
Detailed measurements of several devices are presented in the Main Text in the Supplementary Material notes. For
reference, we provide a look-up table in Tab. S-1 for each of these devices and their measured optical and mechanical
properties.
7
IV. MATERIALS AND METHODS: OPTICAL-ABSORPTION-INDUCED BATH
A. Theoretical Model of the Bath
Optical absorption is found to induce additional parasitic heating and damping of the high-
Q
acoustic breathing
mode of the Si OMC devices at millikelvin temperatures. This absorption heating is thought to proceed through
excitation of sub-bandgap electronic defect states at the Si surfaces which undergo phonon-assisted decay, generating
a local bath of thermal phonons coupled to the high-
Q
breathing mode [33]. We may gain some understanding of the
optically-induced bath by considering a simple model of phonon-phonon interactions which can couple the optically-
induced hot phonon bath to the breathing mode. As we are concerned in this work with the phonon dynamics at low
bath temperature (
T
b
.
10 K), and the acoustic mode of interest is at microwave frequencies, the phonon-phonon
interactions leading to heating and damping of the breathing mode can be understood in terms of a Landau-Rumer
scattering process [13, 34] (see App. VI). In this context, we may consider a simple model in which our mode of interest
at frequency
ω
m
is coupled to higher-frequency bath phonon modes at frequencies
ω
1
and
ω
2
, with
ω
2
ω
1
=
ω
m
. Then
we may write the scattering rates into and out of the mode of interest to first order in perturbation theory [13, 33] as
Γ
+
=
A
(
n
m
+ 1)(
n
2
+ 1)
n
1
and Γ
=
An
m
n
2
(
n
1
+ 1), respectively, where
n
1
,
n
2
, and
n
m
are the number of phonons
in each mode involved in the scattering and
A
is a constant describing the Si lattice anharmonicity. Then the overall
rate of change in the occupancy of the mode of interest is,
̇
n
m
= Γ
+
Γ
=
A
(
n
1
n
2
)
n
m
+
An
2
(
n
1
+ 1)
.
(S-4)
This expression has exactly the form of a harmonic oscillator coupled to a thermal bath with rate
γ
p
=
A
(
n
1
n
2
)
and effective occupancy
n
p
=
An
2
(
n
1
+ 1)
p
. Assuming thermal occupancies for each of the higher frequency phonon
modes of the hot bath,
n
1
,
2
=
n
B
[
~
ω
1
,
2
/k
B
T
p
]
1
/
(exp [
~
ω
1
,
2
/k
B
T
]
1), and using the identity
n
B
[
x
+
x
](
n
B
[
x
]+1) =
(
n
B
[
x
]
n
B
[
x
+
x
])
n
B
[
x
] [13], one finds that the mode
m
thermalizes with the hot bath via 3-phonon scattering
to an effective occupancy which is
n
p
=
n
B
[
~
ω
m
/k
B
T
p
]. This result holds when the hot bath thermalizes to some
temperature independent of the interactions with mode
m
.
In the real material system of the nanobeam, the local hot phonon bath at elevated temperature
T
b
is expected to
be generated as electronic states at
eV energy undergo phonon-assisted relaxation processes, emitting a shower of
high-frequency phonons which subsequently decay by a cascade of nonlinear multi-phonon interactions into a bath
of GHz phonons. Due to the geometric aspect ratio of the thin-film nanobeam, 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. The beam thickness (
t
= 220 nm, width
w
560 nm,
length
l
15
μ
m) corresponds to a relatively high cutoff frequency in the vicinity of
ω
co
/
2
π
v
l
/
(2
t
)
20 GHz,
where
v
l
= 8
.
433 km/s is the longitudinal-phonon velocity in Si. This cutoff frequency imposes an effective phonon
bottleneck
preventing further rapid thermalization to lower-lying modes and a resulting buildup in the bath phonon
population above the bottleneck. For phonon frequencies below the cutoff, where the wavelength is large enough
to approach the lattice constant of the acoustic bandgap clamping region, the reflectivity of the clamping region
increases as ballistic radiation out of the nanobeam is suppressed. The result is a reduced density of phonon states
near and below the cutoff, where the nanobeam supports quasi-discrete (and long-lived, especially in the vicinity of
the mirror bandgap and acoustic shield bandgap) phonon modes at lower frequency as outlined in Fig. S-5. The
phenomenological coupling rate
γ
p
describes the rate at which the lower-lying modes—in particular the breathing
mode at 5 GHz—are coupled to the elevated-temperature bath of higher-frequency phonons above the bottleneck.
In the context of this proposed phonon-bottleneck model, we now consider instead of a discrete pair of modes
n
1
and
n
2
a quasi-continuum of high-frequency bath modes coupled to the mode of interest via some anharmonicity
matrix element
A
(
ω
;
ω
m
). We will assume that the thermal phonons populating the bath have sufficient time to
thermalize amongst each other before decaying, or in other words, that they couple to each other at a mixing rate
γ
mix
much greater than their coupling rates to the external environment or to the lower-lying phonon modes. Under
this assumption, we may define an effective local temperature
T
p
such that the occupancy of a bath phonon at
frequency
ω
is given by the Bose-Einstein occupation factor
n
b
[
ω
;
T
p
]
n
B
[
~
(
ω
ω
co
)
/k
B
T
p
] =
1
e
~
(
ω
ω
co
)
/k
B
T
p
1
,
(S-5)
where
ω
co
represents the new effective ground-state frequency due to the phonon bottleneck effect, and
n
B
[
x
] =
1
/
(exp[
x
]
1) is the Bose distribution.
The temperature of the optically-induced hot phonon bath,
T
p
, can then be related to the absorbed optical power
P
abs
using a model of the lattice thermal conductivity in the low temperature limit. Assuming the optical absorption
8
process is linear, we can write the absorbed optical power as a fraction
η
of the optical pump power:
P
abs
=
ηP
in
=
η
n
c
.
In steady state, the power output into the phonon bath is equal to its input,
P
out
=
P
abs
n
c
. The lattice thermal
conductivity at low temperatures, where phonon transport is ballistic, scales as a power law of the phonon bath
temperature [35, 36],
G
th
(
T
p
)
α
. 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. In the case of a structure with 2D phonon density of states,
such as the OMC cavity in the frequency range from 10-100 GHz (c.f., Fig. S-15),
α
=
d
= 2 and the hot phonon
bath temperature scales as
T
p
P
1
/
3
out
n
1
/
3
c
. This approximate scaling is expected to be valid so long as phonons
in the hot phonon bath approximately thermalize each other upon creation from optical absorption events, and then
radiate freely (ballistically) into the effective zero temperature substrate. The picture one has then is that the hot bath
phonons make multiple passes within the OMC cavity region, scattering with other phonons leading to thermalization,
and then eventually radiating into the substrate, i.e., the OMC cavity is still a good cavity for many phonons in the
acoustic frequency region above the phononic bandgap.
In analogy with Equation S-4, for a phonon bath density of states
ρ
(
ω
) we can calculate the effective coupling rate
γ
p
between the hot phonon bath and the mode of interest due to 3-phonon scattering:
γ
p
=
ˆ
ω
co
d
ω A
[
ω
;
ω
m
]
ρ
[
ω
]
ρ
[
ω
+
ω
m
] (
n
b
[
ω
]
n
b
[
ω
+
ω
m
])
,
(S-6)
In a simple continuum elastic model [13, 33], the product of the anharmonicity matrix element
A
[
ω
;
ω
m
] and the
density of states is taken to obey a polynomial scaling
A
[
ω
;
ω
m
]
ρ
[
ω
]
ρ
[
ω
+
ω
m
] =
A
(
ω
ω
co
)
a
for some constants
A
and
a
, where we have introduced the cut-off frequency below which we assume the density of states is zero. With this
assumption,
γ
p
=
A
ˆ
ω
co
d
ω
(
ω
ω
co
)
a
(
n
b
[
ω
]
n
b
[
ω
+
ω
m
])
(S-7)
=
A
ˆ
ω
co
d
ω
(
ω
ω
co
)
a
(
n
b
[
ω
+
ω
m
](
n
b
[
ω
] + 1)
n
B
[
~
ω
m
/k
B
T
p
]
)
(S-8)
=
A
n
B
[
~
ω
m
/k
B
T
p
] + 1
ˆ
ω
co
d
ω
(
ω
ω
co
)
a
(
n
b
[
ω
](
n
b
[
ω
+
ω
m
] + 1))
,
(S-9)
where in the last line we used the identity
n
B
[
x
+
x
](
n
B
[
x
] + 1)
/n
B
[
x
] = (
n
B
[
x
+
x
] + 1)
n
B
[
x
]
/
(
n
B
[
x
] + 1). Making
a change of variables to
x
~
(
ω
ω
co
)
/k
B
T
p
in the integral in Eq. (S-9), we have
γ
p
=
(
A
n
B
[
x
m
] + 1
)(
k
B
T
p
~
)
a
+1
ˆ
0
d
x x
a
(
n
B
[
x
](
n
B
[
x
+
x
m
] + 1))
(S-10)
where
x
m
=
~
ω
m
/k
B
T
p
. The integral in Equation S-10 depends on temperature only through
x
m
, and in the small
and large
x
m
limit (corresponding to low and high temperature), is relatively independent of
x
m
. If we assume that
the anharmonicity element
A
[
ω
;
ω
m
] is approximately frequency independent, and the only frequency dependence in
A
(
ω
ω
co
)
a
comes from the phonon density of states, then
a
2(
d
1) for a phonon bath of dimension
d
. We can
thus make a general observation about the scaling of the bath-induced damping rate
γ
p
in the low (
x
m

1) and high
(
x
m

1) temperature regimes:
γ
p
{
(
k
B
T
p
~
)
a
n
2(
d
1)
/
(
d
+1)
c
for
T
p

~
ω
m
k
B
,
(
k
B
T
p
~
)
a
+1
n
(2
d
1)
/
(
d
+1)
c
for
T
p

~
ω
m
k
B
,
(S-11)
for a generic hot phonon bath of dimension
d
. In a structure such as the OMC nanobeam cavity we expect the
dimensionality of the effective bath density of states to be reduced relative to the Debye 3D density of states for a
bulk crystal. Here we will assume - consistent with numerical simulations of the OMC structure in Section VI - that
the phonon bath has a two-dimensional density of states corresponding to
a
= 2. In this case, we have the following
scaling of the damping factor with intra-cavity photon number,
γ
p
{
(
k
B
T
p
~
)
2
n
2
/
3
c
for
T
p

~
ω
m
k
B
,
(
k
B
T
p
~
)
3
n
c
for
T
p

~
ω
m
k
B
.
(S-12)
9
Upon thermalizing with the hot phonon bath, the effective thermal occupancy
n
p
of the high-
Q
breathing mode
of the acoustic cavity can be found from a similar rate equation analysis as considered for the 3-mode scattering in
Eq. (S-4). Integrating over all the possible 3-phonon scattering events involving the mode of interest at frequency
ω
m
yields,
n
p
=
1
γ
p
ˆ
ω
co
d
ω A
[
ω
;
ω
m
]
ρ
[
ω
]
ρ
[
ω
+
ω
m
]
n
b
[
ω
+
ω
m
] (
n
b
[
ω
] + 1])
(S-13)
=
n
b
[
ω
co
+
ω
m
]
A
γ
p
ˆ
ω
co
d
ω ω
a
(
n
b
[
ω
]
n
b
[
ω
+
ω
m
])
(S-14)
=
n
B
[
~
ω
m
/k
B
T
p
]
.
(S-15)
We therefore have a characteristic scaling behavior for the effective phonon occupancy
n
p
coupled to the cavity mode
of interest that is,
n
p
{
(
k
B
T
p
~
ω
m
)
n
1
/
(
d
+1)
c
d=2
=
n
1
/
3
c
for
T
p

~
ω
m
k
B
,
exp [
~
ω
m
/k
B
T
p
]
for
T
p

~
ω
m
k
B
.
(S-16)
B. Measurement of optical-absorption-induced damping,
γ
p
In order to measure the additional bath-induced damping rate
γ
p
, we use a pump-probe technique employing two
laser sources. The pump laser is tuned to optical resonance (∆ = 0) to eliminate dynamical back-action effects
(
γ
OM
= 0), and impinges on the cavity in continuous-wave (CW) operation. The pump laser generates a steady-state
intracavity photon population
n
c,CW
and an absorption-induced bath at elevated temperature in the steady state. A
second pulsed laser, the probe laser, is tuned to the red motional sideband of the cavity (∆ = +
ω
m
) and is used to
periodically read out the phonon occupancy, where the scattering rate of the probe laser at the beginning of the probe
pulse provides an estimate of
n
p
due to the CW laser alone. Application of the probe laser not only allows readout
of the breathing mode occupancy, but also produces an excess absorption-induced bath above and beyond that of the
background CW laser alone. When the readout probe pulse is turned off, the breathing mode initially heats due to
the excess hot bath created by the probe pulse (over several microseconds; see Fig. S-7(c)), and then after this excess
hot bath evaporates away leaving a breathing mode occupancy of
n
f
, relaxes back to its steady-state occupancy set
by the CW laser,
n
m
[
n
c,CW
] = (
γ
p
[
n
c,CW
]
n
p
[
n
c,CW
] +
γ
0
n
0
)
/
(
γ
p
[
n
c,CW
] +
γ
0
). The rate of relaxation is set by the
modified total damping rate of
γ
0
+
γ
p
[
n
c,CW
]. By observing this modified exponential decay rate we directly extract
γ
p
[
n
c,CW
], with
γ
0
known from independent ringdown measurements in the absence of the CW background laser. For
example, in Fig. S-6(a) we show the measured ringdown of a high-
Q
six-shield device (device B;
γ
0
/
2
π
= 0
.
21 Hz) for
a CW pump laser photon number of
n
c,CW
= 10
2
, from which we extract
γ
p
/
2
π
= 42
.
8 Hz.
For large
n
c,CW
(
&
1) the steady-state occupancy of
n
m
[
n
c,CW
] becomes larger than the occupancy ̃
n
f
m
at the end
the readout pulse. The readout pulse
should
cool the breathing mode, after all, and it is only the absorption-induced
heating caused by the readout pulse itself that manifests as a ring down in absence of heating from the CW laser.
For large
n
c,CW
then,
γ
p
is estimated by observing a ring-
up
in the pulse-off state from the final pulse occupancy ̃
n
f
m
to the elevated
n
m
[
n
c,CW
]. Figure S-6(b) shows a representative data set for extracting
γ
p
at
n
c,CW
>
1, where an
initial fast rise is observed in the mode occupancy in the pulse-off state from
n
f
m
to ̃
n
f
m
due to the aforementioned
excess bath created by the readout pulse, followed by a slower second heating stage from ̃
n
f
m
to
n
c,CW
. As discussed
in more detail in sub-Section IV D, we can fit the ring up curve after the readout pulse is turned off by considering a
phenomenological model including decay of the readout-induced hot bath,
̇
n
m
=
{
γ
0
+
γ
p
[
n
c,RO
]
e
ζ
γ
p
τ
+
γ
p
[
n
c,CW
]
}
n
m
+
{
γ
p
[
n
c,RO
]
e
ζ
γ
p
τ
+
γ
p
[
n
c,CW
]
}
(
n
p
[
n
c,RO
]
e
ζ
n
p
τ
+
n
p
[
n
c,CW
])
.
(S-17)
We first measure the transient readout-induced bath in the absence of the CW laser (dark green curve in Fig. S-6b),
from which a fit to Eq. (S-17) yields
n
p
[
n
c,RO
] = 40 phonons,
γ
p
[
n
c,RO
]
/
2
π
= 9
.
55 kHz,
ζ
γ
p
/
2
π
= 143 kHz, and
ζ
n
p
/
2
π
= 15
.
9 kHz. With these readout-induced bath values known, Eq. (S-17) is numerically integrated to fit the
entire heating curve in the pulse-off state to extract the additional CW-pump-induced damping
γ
p
[
n
c,CW
].
The results of the measured optical-absorption-induced damping
γ
p
versus
n
c
are summarized in Fig. S-7(a) for
measurements on both a six-shield (device B) and a zero-shield (device A) nanobeam device. The observed power law
10
scaling fits well to
γ
p
/
2
π
= (1
.
07 kHz)
×
n
2
/
3
c
, in agreement with the scaling predicted in Eq. (S-12) for a 2D density
of states for the bath phonon population. Note that the much lower
γ
0
of the six-shield device allows a much wider
range of
γ
p
(and thus
n
c
) to be explored.
C. Measurement of optical-absorption-induced bath occupancy,
n
p
In order to measure the bath occupancy
n
p
, again two different methods are used to probe the high- and low-
photon-number dependencies of the bath. To measure the bath occupancy at photon numbers
n
c
&
1, a simple
readout technique may be used in which a single readout laser is sent to the cavity in continuous-wave operation. The
laser is tuned to cavity resonance (∆ = 0) and the resulting sideband scattered photon count rate appearing at either
the lower or upper frequency mechanical sideband (∆ =
±
ω
m
) will be
Γ = Γ
noise
+
(
κ
2
ω
m
)
2
Γ
SB,0
n
m
.
(S-18)
With the sideband filters aligned to either of the mechanical sidebands of the cavity, the observed count rate is used
to extract an equivalent occupancy
n
m
=
n
p
at various pump powers
n
c
. The results are shown in Fig. S-7b (orange
circles) for a six-shield device (device B), exhibiting a power-law scaling of
n
p
= (7
.
94)
×
n
1
/
3
c
in agreement with
the model in the limit of high bath temperature
T
p

~
ω
m
/k
B
200 mK. The right-hand axis of Fig. S-7 gives the
effective bath temperature
T
p
corresponding to the measured occupancy, indicating that the measurement regime is
indeed well in the high temperature limit.
At lower photon numbers
n
c
.
1, and corresponding lower
n
p
n
m
, the SNR of the counting of photons scattered
from cavity resonance into either mechanical sideband begins to drop below 1 due to the large sideband resolution
factor (2
ω
m
)
2
of the OMC cavity (c.f., Eq. (S-18)). In this regime, an alternative measurement method is employed
in which a CW pump laser generates a steady-state optical-absorption bath while a second pulsed readout laser is
used to probe the breathing mode occupancy (see Fig. S-8). The background pump laser is detuned to ∆
/
2
π
= 1 GHz
from the cavity resonance to minimize back-action as well as bleed-through counts through the sideband filters aligned
at ∆ = 0. The initial measured occupancy
n
i
m
during the pulse is a measure of the pump-induced bath occupancy;
however, it includes a small residual occupancy ̃
n
0
0
.
04 due to heating caused by the readout laser prior to the
first measurement time bin of the pulse-on state. We define a corrected occupancy
n
m
n
i
m
̃
n
0
which denotes the
measured mode occupancy which is coupled to the fridge bath as well as the absorption-bath induced by the pump
laser:
n
m
=
n
p
γ
p
+
n
0
γ
0
γ
p
+
γ
0
.
(S-19)
With
n
0
,
γ
0
, and the power-dependence of
γ
p
known from independent measurements, we can estimate the equivalent
bath occupancy
n
p
[
n
c
] =
n
m
γ
p
[
n
c
] + (
n
m
n
0
)
γ
0
γ
p
[
n
c
]
.
(S-20)
Using this second method, over a much larger span of
n
c
, the behavior of the effective bath occupancy
n
p
for a
zero-shield device with intrinsic damping rate
γ
0
/
2
π
= 14
.
1 kHz is shown in Fig. S-7b as purple squares. Note that
measurement of the very high-
Q
six-shield device (device B) using the pulsed readout scheme is not practical due to
the extremely long relaxation times required between readout pulses (we did, however, verify for a few values of
n
c
that the two schemes give consistent results). For
n
c
&
1, again we find good agreement for the zero-shield device
with a power-law scaling
n
p
n
1
/
3
c
for
T
b

~
ω
m
/k
B
. Not only is the scaling of
n
p
versus
n
c
the same for both
zero-shield and six-shield devices, but so is the absolute value of
n
p
. For
n
c
.
1,
γ
p
(
n
c
)
γ
0
for the zero-shield device
and the measured occupancy
n
m
deviates substantially from
n
p
as the breathing mode thermalizes more strongly with
the external substrate temperature set by the fridge (
T
f
10 mK). In this range we have plotted
n
m
in translucent
purple squares to distinguish it from the region of parameter space where
n
m
is expected to faithfully represent
n
p
.
11
D. Measurement of optical-absorption-induced bath dynamics
The hot bath created by the application of laser light resonant with the optical mode of the OMC cavity does not
instantaneously appear when the laser light is turned on, nor does it instantaneously vanish once the laser is turned
off. Rather, there is a somewhat complicated bath dynamics that can be inferred from careful study of the temporal
variation of the scattered photon signal from the readout pulse due to excitation of the mechanical breathing mode by
the hot phonon bath. Using the measured breathing mode occupancy as a proxy one can infer many subtle features
of the bath dynamics.
Figure S-9(a) shows the measured scattered photon signal due to a pulsed readout tone applied on the lower
motional sideband of the optical cavity (∆ = +
ω
m
) of a high mechanical
Q
-factor six-shield OMC device (device B).
Here the readout pulses are
T
pulse
= 4
μ
s long and a variable delay
τ
is applied between each successive optical readout
pulse. The scattered photons from the readout pulse are filtered by the filter bank resonantly aligned with the optical
cavity resonance (∆ = 0), thus yielding a photon count rate throughout the readout pulse which is proportional to
the average occupancy of the mechanical breathing mode
n
m
. This is shown in Fig. S-9(a) for a time bin resolution
of 10
.
24 ns, with the first measurement bin occurring at
t
= 100 ns after the pulse-on signal is applied in order to
ensure that the optical pulse amplitude has settled and reached its maximum value. In the left panel of Fig. S-9(b)
we plot time-varying normalized breathing mode occupancy, corresponding to the ratio of the measured signal during
the pulse to that at the very end of the pulse. This curve is not a single-shot measurement, but rather averaged over
thousands of pulses, for which the normalized signal avoids small, slow drifts in the efficiency of the measurement
apparatus. In the right panel we plot the normalized initial measurement bin occupancy (still taken to be at 100 ns
after the readout pulse is turned on) as a function of the off-state delay time
τ
between successive pulses.
Several things are quickly evident from these plots of the measured breathing mode occupancy during and after
the applied optical pulse. During the pulse we expect the optomechanical back-action to induce damping and cooling
of the breathing mode at a rate
γ
OM
[
n
c
]. Without any parasitic heating effects from the applied optical pulse, the
breathing mode should cool down to its equilibrium occupancy, ideally very close to zero at the fridge temperature
(
T
f
= 10 mK). This does not occur, but rather the breathing mode occupancy is seen to initially cool to a few phonons
over
300 ns, and then slowly heat to a steady-state phonon occupancy at the end of the pulse of
n
f
m
= 4
.
2 phonons
(c.f., Fig. S-9(a)). Similarly, once the optical pulse is turned off and light has left the optical cavity, the breathing
mode occupancy starts to heat again, levelling off after a few microseconds following a slight overshoot to a modified
post-pulse value of
n
m
[0] = 27 phonons (
n
i
m
= 13
.
6 phonons in the first masurement bin; c.f., Fig. S-7(b)). This
strange dynamics is a result of the coupling of the breathing mode to the optical-absorption-induced hot bath. The
slight undershoot of the cooling and slow heating in the pulse-on state is a result of a slow turn on of the hot bath.
Similarly, the transient post-optical-pulse heating results from the slow decay of the hot bath, now without the cooling
from optomechanical back-action.
Noticeably, the timescales for the turn on (
400 ns) of the bath and the turn off (
3
μ
s) of the bath are different.
Less evident from these plots, but nonetheless very clear when attempting to model the hot bath dynamics, is that
there seems to be two components to the bath, one whose turn on and turn off transients are very rapid (effectively
instantaneous with the optical field), and one with much slower relaxation times. Even more subtle is that to get very
good agreement with the measured initial transient in the breathing mode occupancy in the immediate aftermath
of turning off the optical pulse, it seems that the hot-bath damping factor,
γ
p
, should be modeled with a more
rapid relaxation rate than that of the hot bath occupancy,
n
p
. It may be that this is also the case in the transient
dynamics during the pulse-on state, however, in the pulse-off state the relaxation rate of the measured breathing mode
occupancy is far more sensitive to the value of
γ
p
as it dominates the total relaxation rate of the breathing mode in
the absence of appreciable
γ
OM
.
The model used to fit the data in Fig. S-9 consists of a set of coupled differential equations involving the breathing
mode occupancy, the hot bath damping factor, and the effective hot bath occupancy. The rate equation for the
breathing mode occupancy is given by,
̇
n
m
=
(
γ
p
+
γ
OM
+
γ
0
)
n
m
+
γ
p
n
p
+
γ
0
n
0
(S-21)
where in the pulse-on state
γ
OM
=
γ
OM
[
n
c
] will take on a large value on the order of 1 MHz for a readout pulse
amplitude of a few hundred intra-cavity photons, and in the pulse-off state
γ
OM
0 due to the large extinction
(
&
80 dB) and rapid timescale of the turn-off the optical pulse ( 20 ns). During the pulse-on state the rate equations
for the fast (F) and slow (S) components of the hot bath damping factor and effective occupancy are,
̇
(
γ
p
)
F(S)
(
t
) =
(
θ
γ
p
)
F(S)
{
(
γ
p
)
F(S)
(
t
)
(
δ
b
)
F(S)
γ
p
[
n
c,RO
]
}
,
(S-22)
12
and
̇
(
n
p
)
F(S)
(
t
) =
(
θ
n
p
)
F(S)
{
(
n
p
)
F(S)
(
t
)
(
δ
b
)
F(S)
n
p
[
n
c,RO
]
}
,
(S-23)
where
t
=
{
0
,T
pulse
}
is the time from the start of the pulse to the end of the pulse, (
θ
γ
p
)
F(S)
and (
θ
γ
p
)
F(S)
are
the pulse-on relaxation rate constants for the damping factor and occupancy of the two different bath components,
respectively, and (
δ
b
)
F(S)
is the F(S) fraction of the hot bath.
γ
p
[
n
c,RO
] and
n
p
[
n
c,RO
] are the steady-state bath values
reached at the end of the optical readout pulse. The corresponding rate equations for the hot bath in the pulse-off
state are,
̇
(
γ
p
)
F(S)
(
τ
) =
(
ζ
γ
p
)
F(S)
{
(
γ
p
)
F(S)
(
τ
)
(
δ
b
)
F(S)
γ
p
[
n
c,RO
]
}
,
(S-24)
and
̇
(
n
p
)
F(S)
(
τ
) =
(
ζ
n
p
)
F(S)
{
(
n
p
)
F(S)
(
τ
)
(
δ
b
)
F(S)
n
p
[
n
c,RO
]
}
.
(S-25)
where
τ
is the time from the end of the optical pulse, and (
ζ
γ
p
)
F(S)
and (
ζ
γ
p
)
F(S)
are the pulse-off relaxation rate
constants for the damping factor and occupancy of the two different bath components, respectively.
The model parameters used to fit the specific measured data for the six-shield device (device B) presented in
Fig. S-9 are listed in Table S-2. Similar bath dynamical parameters are found for all of the measured devices we have
studied. Independent of the optical readout pulse power, the fraction of the bath which reacts quickly seems to be
consistently close to a value of (
δ
b
)
F
= 0
.
65. The fast component of the bath turns on faster than we can resolve
(
&
50 MHz), while slow component of the bath turns on with a rate constant of approximately
θ
S
/
2
π
= 600 kHz (for
both damping factor and occupancy). The fast component of the bath turns off with an exponential rate constant
of (
ζ
γ
p
)
F
/
2
π
= 150 kHz for
γ
p
and (
ζ
n
p
)
F
/
2
π
= 70 kHz for
n
p
. Even more slowly, the slow component of the bath
turns off with a rate constant of (
ζ
γ
p
)
S
/
2
π
= 90 kHz and (
ζ
n
p
)
S
/
2
π
= 24 kHz for the two different bath factors.
Our ability to measure the bare damping rate of the acoustic breathing mode relies on the fact that the hot bath
evaporates prior to the actual measurement of the free decay of the breathing mode. This means that the first
10
μ
s
of the pulse-off state is dead time in which the dynamics of the breathing mode occupancy is still coupled to that of
the hot bath. Crucial to the measurement of a ringdown curve using the red-detuned optical pulse as both a readout
signal and an excitation source, is that after this dead time there remain a residual, elevated phonon occupancy of
the breathing mode from which the mode can decay. This is clearly the case for the data measured in Fig. S-9, and is
a result of the fact that at this readout power the peak magnitude of
γ
p
(2
π
(85 kHz)) is still smaller than the fastest
decay of the hot bath ((
ζ
γ
p
)
F
/
2
π
= 150 kHz), so that the breathing mode occupancy cannot follow that of the fast
dynamics of the hot bath. This non-adiabatic quenching leaves the breathing mode with an elevated occupancy after
the dead time. At readout powers beyond
n
c
= 1000 this stops being the case, hence our choice of readout pulse
powers
n
c
.
600 in the ringdown measurements.
A few observational comments are warranted. The fact that the hot bath should have faster pulse-on rate constants
than pulse-off rate constants might be explained by the fact that there are likely a wide spectrum of phonons which
are created by absorption of the optical pulse. This may lead to a hierarchy of phonon baths. Consider for instance
a two bath scenario, consisting of a high and a low frequency phonon bath. The high frequency bath is assumed to
be directly populated from optical absorption events, while the low frequency bath is predominantly responsible for
coupling to the breathing mode of interest. In the high frequency phonon bath, phonons rapidly mix with each other
due the large density of states and mode occupancy. The high frequency phonon bath is also well thermalized to the
external substrate through acoustic radiation. Phonons in the low frequency bath are fed from the phonon-phonon
scattering processes in the high frequency phonon bath, and are less connected via radiation to the substrate. When
the optical pulse is on, the high frequency bath is rapidly populated. The high frequency bath not only acts as a
source of phonons for the low frequency bath, but through nonlinear phonon mixing also helps bring it into some
quasi-equilibrium temperature. When the optical pulse is turned off, the high frequency bath rapidly decays away,
leaving the low frequency bath of phonons to more slowly decay away due to the absence of the phonons in the high
frequency bath to mix with. This scenario would also explain the difference in the decay of the low frequency bath
γ
p
damping rate, which depends on the phonon number density, to that of the effective occupancy
n
p
, which is set by
the quasi-equilibrium temperature of the bath. The absence of the high frequency bath could greatly slow down the
low frequency bath equilibriation rate, and thus the rate of change of the effective bath temperature, while the low
frequency bath acoustic coupling to the external substrate will provide a constant decay channel for bath phonons
and thus
γ
p
.
13
We should further note that the dynamical bath parameters reported in Table S-2 are consistent for devices fabri-
cated from low resistivity SOI. In the case of our high resistivity SOI samples, we have measured devices with much
slower post-read-pulse decay of the hot bath. Hot bath decay times as long as tens of milliseconds have been observed.
Although requiring further study, we believe that this very slow decay dynamics of the optical-absorption-induced
bath indicate that phonons are not the only parties involved in the optically-induced hot bath, but that the hot bath is
likely also composed of much longer lifetime two-level system (TLS) defects. The high resistivity SOI seems to harbor
much longer-lived TLS states, possibly due to the reduction of electronic relaxation pathways. Our two-phonon-bath
scenario above may in fact be a two-bath scenario consisting of one phonon bath coupled to a longer lived TLS bath.
Other evidence for this interpretation is the high values of measured
γ
p
which is more consistent with estimated TLS
damping rates (damping due to three-phonon scattering is shown to be too slow, at least for bath temperatures below
1 K, in sub-Section VIII A).
14
V. MATERIALS AND METHODS: COHERENT EXCITATION METHODS
A. Low-Threshold Acoustic Self-Oscillation
Owing to the extremely slow intrinsic damping rate
γ
0
observed in the ultra-high-
Q
nanobeam devices at low
temperature, it is possible to drive the mechanics into the regime of self-sustained oscillations with a blue-detuned
pumping laser at very low input optical powers, or equivalently, a very low rate of measurement back-action. The total
effective damping rate experienced by the mechanics in the presence of a blue-detuned drive laser is
γ
=
γ
i
γ
OM
,
where the intrinsic damping rate
γ
i
=
γ
0
+
γ
p
includes damping
γ
0
from both the cold fridge bath (with occupancy
n
0
10
3
) and from the optical absorption-induced phonon bath at rate
γ
p
. The usual condition for self-oscillation
is that the damping rate is matched by the back-action amplification rate
γ
OM
:
γ
OM
=
γ
0
+
γ
p
.
(S-26)
We observe the onset of mechanical self-oscillation at
T
f
= 10 mK, in which a CW blue-detuned pump laser drives
the cavity and the sideband filters are aligned to the cavity resonance (∆ = 0). The scattered photon count rate Γ
SB,0
is measured in steady-state. In the setup configuration used for these measurements, an additional VOA is placed
in the optical path, elevating the measured SPD dark count rate to 10
.
8 c.p.s. Sweeping the input power (photon
number)
n
c
results in a sharp increase in detected count rate at the self-oscillation threshold
n
c,thresh
= 2
×
10
3
as
shown in Figure S-10, where we estimate the resulting steady-state phonon occupancy to be of order
n
m
5
×
10
4
.
At the threshold
n
c,thresh
we can estimate the back-action amplification rate
γ
OM
/
2
π
= 4
g
2
0
n
c,thresh
8 Hz from
the known optical device parameters, indicating that the intrinsic damping
γ
i
is dominated by the bath damping rate:
γ
p
(
n
c,thresh
) =
γ
OM
(
n
c,thresh
)
γ
0
2
π
(7
.
9 Hz), in good quantitative agreement with the trend measured on a similar
device in Figure S-7a. Upon decreasing the driving power (green data in Figure S-10), self-oscillation appears to relax
at a decreased threshold of
n
c
= 1
.
4
×
10
3
, indicating a hysteresis in the measured count rates as a function of input
power. This apparent hysteresis likely arises from a change in the
true
intracavity photon number as a function of
driving power
P
in
. We have so far adhered to Equation S-1 in determining the
n
c
as a function of
P
in
; this expression
is used to generate the horizontal axis of Figure S-10, and so does not represent the true intracavity photon number.
However, in order to unambiguously calculate
n
c
in the presence of large phonon amplitude
n
m
, a more thorough
calculation is needed which accounts for the effective optical reflection profile in the presence of strong modulation by
the mechanical motion.
B. Electromagnetically Induced Transparency Mechanical Spectroscopy
Electromagnetically induced transparency (EIT) in optomechanical systems allows for a spectral measurement of
the mechanical response via observation of a transparency window in the optical reflection spectrum. A pump laser
tone at
ω
l
is amplitude modulated to generate a weak probe tone at
ω
s
,
±
=
ω
l
±
p
. If the pump-cavity detuning is
fixed on either the red- or blue-side of the optical cavity (∆ =
±
ω
m
), the optical susceptibility of the cavity strongly
suppresses one of the probe sidebands (at
ω
s
,
) and only the other probe sideband will have an appreciable intracavity
population. For a red-detuned pump, the interaction of the pump tone and mechanics with the probe sideband yields
a reflection coefficienct
r
(∆
) for the probe which contains a transparency window having a width on the scale of
the mechanical mode linewidth:
r
(∆
) = 1
κ
e
κ/
2 +
i
(∆
(
δ
+
ω
m
)) +
|
G
|
2
+
γ
i
/
2
,
(S-27)
where we have defined
δ
p
ω
m
and
G
g
0
n
c
. We measure the reflection amplitude
R
=
|
r
|
2
by driving an
EOM weakly to generate a probe tone and observing the count rates of sideband-scattered probe photons. The pump
is locked at ∆ = +
ω
m
and the cascaded filter stack is locked to the cavity frequency. The RF modulation power is
chosen to generate a sideband intracavity photon number much smaller than the carrier photon number (
n
c
,+

n
c
)
while maintaining a large count rate
10
5
c.p.s. at the SPDs to minimize data integration times. This corresponds
to modulation indices in the range of
β
10
3
for our system parameters (measurements were performed on device
D). The modulation frequency ∆
p
is swept over a range of about 1 MHz to map out the transparency window. This
range is large enough to include the optomechanically-broadened mechanical linewidth which sets the width of the
transparency window, but much narrower than the bandwidth of the FFP filters (
50 MHz), allowing for the filters to