of 12
www.sciencemag.org/content/34
9/6251/952
/suppl/DC1
Supplementary
Material
s for
Quantum squeezing of motion in a mechanical resonator
E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald,
F. Marquardt, A. A. Clerk, K. C. Schwab*
*C
orrespond
ing author
. E
-mail: schwab@caltech.edu
Published
28
August
201
5,
Science
349
, 952
(201
5)
DOI:
10.1126/science.
aac5138
This PDF file includes:
Materials and Methods
Figs. S1 to S5
1 Theory
1.1 Linearized optomechanical Hamiltonian and quantum Langevin equa-
tions
In this section, we establish the equations of motion for our system using input-output theory. The
Hamiltonian of a generic optomechanical system reads
ˆ
H
=
~
ω
c
ˆ
a
ˆ
a
+
~
ω
m
ˆ
b
ˆ
b
~
g
0
ˆ
a
ˆ
a
(
ˆ
b
+
ˆ
b
)
+
ˆ
H
drive
,
(S.1)
where
ˆ
a
(
ˆ
a
)
is the annihilation (creation) operator of the intra-cavity field,
ˆ
b
(
ˆ
b
)
is the mechanical
phonon annihilation (creation) operator, and
g
0
is the bare optomechanical coupling between the
cavity and the mechanical oscillator.
ˆ
H
drive
describes the external driving.
In our experiment, the optomechanical system consists of an electromechanical system with
two ports. We drive the cavity mode with two microwave tones from the left port, which we
designate
(
L
)
. The drive Hamiltonian reads
ˆ
H
drive
=
~
ν
=
±
α
ν
(
ˆ
ae
ν
t
+ ˆ
a
e
ν
t
)
,
(S.2)
where
ω
±
=
ω
c
+ ∆
±
(
ω
m
+
δ
)
and
α
±
are the blue and red pump amplitudes at the input port.
The detunings
δ
and
are shown in Fig. SS1. In the following, we apply standard linearization
– i.e., we separate the cavity and the mechanical operators,
ˆ
a
and
ˆ
b
, into a classical part,
̄
a
or
̄
b
,
plus quantum fluctuations,
ˆ
d
or
ˆ
c
. E.g.,
ˆ
a
̄
a
+
ˆ
d
. In the interaction picture with respect to
ˆ
H
0
=
~
(
ω
c
+ ∆) ˆ
a
ˆ
a
+
~
(
ω
m
+
δ
)
ˆ
b
ˆ
b
, we find the
linearized
optomechanical Hamiltonian
ˆ
H
lin
=
ˆ
H
RWA
+
ˆ
H
CR
.
(S.3)
Here,
ˆ
H
RWA
=
~
ˆ
d
ˆ
d
~
δ
ˆ
c
ˆ
c
~
[
(
G
+
ˆ
c
+
G
ˆ
c
)
ˆ
d
+
(
G
+
ˆ
c
+
G
ˆ
c
)
ˆ
d
]
(S.4)
describes the resonant part of the linearized optomechanical interaction whereas
ˆ
H
CR
=
~
[
G
+
e
2
i
(
ω
m
+
δ
)
t
ˆ
c
+
G
e
2
i
(
ω
m
+
δ
)
t
ˆ
c
]
ˆ
d
~
[
G
+
e
2
i
(
ω
m
+
δ
)
t
ˆ
c
+
G
e
2
i
(
ω
m
+
δ
)
t
ˆ
c
]
ˆ
d
(S.5)
describes off-resonant optomechanical interactions. Note that
G
±
=
g
0
̄
a
±
describes the driven-
enhanced optomechanical coupling. Here,
̄
a
±
is the intracavity microwave amplitude due to the
red and blue pumps, and we have assumed
̄
a
±
R
for simplicity and without loss of generality.
Let us first consider the good cavity limit
(
ω
m

κ
)
. We will later discuss how to incorporate
corrections due to bad cavity effects in Section 1.4. This allows us to work within the rotating
wave approximation. Thus,
ˆ
H
lin
ˆ
H
RWA
. In that case, the linearized quantum Langevin equations
read
̇
ˆ
d
=
(
κ
2
i
)
ˆ
d
+
i
(
G
ˆ
c
+
G
+
ˆ
c
)
+
κ
ˆ
d
in
,
(S.6)
̇
ˆ
c
=
(
γ
m
2
)
ˆ
c
+
i
(
G
ˆ
d
+
G
+
ˆ
d
)
+
γ
m
ˆ
c
in
.
(S.7)
Here,
ˆ
d
in
=
σ
=
L,R,I
κ
σ
κ
ˆ
d
σ,
in
is the total input noise of the cavity, where
ˆ
d
σ,
in
describes the input
fluctuations to the cavity from channel
σ
with damping rate
κ
σ
.
σ
=
L
and
R
correspond to the left
and right microwave cavity ports, while
σ
=
I
corresponds to internal losses. The noise operator
ˆ
c
in
describes quantum and thermal noise of the mechanical oscillator with intrinsic damping rate
γ
m
. The input field operators satisfy the following commutation relations:
[
ˆ
d
σ,
in
(
t
)
,
ˆ
d
σ
,
in
(
t
)
]
=
δ
σ,σ
δ
(
t
t
)
,
(S.8)
[
ˆ
c
in
(
t
)
,
ˆ
c
in
(
t
)
]
=
δ
(
t
t
)
,
(S.9)
ˆ
d
σ
,
in
(
t
)
ˆ
d
σ,
in
(
t
)
=
n
th
σ
δ
σ,σ
δ
(
t
t
)
,
(S.10)
ˆ
c
in
(
t
) ˆ
c
in
(
t
)
=
n
th
m
δ
(
t
t
)
,
(S.11)
where
n
th
σ
is the photon occupation in port
σ
, and
n
th
m
= 1
/
[exp (
~
ω
m
/k
B
T
)
1]
is the thermal
occupation of the mechanical oscillator. The total occupation of the cavity is the weighted sum of
the contributions from different channels:
n
th
c
=
σ
κ
σ
κ
n
th
σ
.
1.2 Optomechanical output spectrum and mechanical spectrum
In this section, we derive the optomechanical output spectrum and the mechanical quadrature spec-
trum, first within the RWA and later in Sec. 1.3 including bad cavity effects. For this, we solve the
quantum Langevin equations (Eqs. S.6, S.7) in Fourier space. It is convenient to define the vectors
D
=
(
ˆ
d,
ˆ
d
,
ˆ
c,
ˆ
c
)
T
,
D
in
=
(
ˆ
d
in
,
ˆ
d
in
,
ˆ
c
in
,
ˆ
c
in
)
T
and
L
=
diag
(
κ,
κ,
γ
m
,
γ
m
)
. We then find
the following solution to the quantum Langevin equations in frequency space:
ˆ
D
[
ω
] =
χ
[
ω
]
·
L
·
ˆ
D
in
[
ω
]
,
(S.12)
where
χ
[
ω
]
κ
2
i
(
ω
+ ∆)
0
iG
iG
+
0
κ
2
i
(
ω
∆)
iG
+
iG
iG
iG
+
γ
m
2
i
(
ω
+
δ
)
0
iG
+
iG
0
γ
m
2
i
(
ω
δ
)
1
.
(S.13)
We measure the output light spectrum through the undriven (right) cavity port. One finds the output
light field
ˆ
d
R,
out
(
ω
)
using the input-output relation
ˆ
d
σ
,out
(
ω
) =
ˆ
d
σ,
in
(
ω
)
κ
σ
ˆ
d
(
ω
)
. This yields
ˆ
d
R,
out
(
ω
) =
ˆ
d
R,
in
(
ω
)
κ
R
κ
(
χ
[
ω
])
11
ˆ
d
in
κ
R
κ
(
χ
[
ω
])
12
ˆ
d
in
(S.14)
κ
R
γ
m
(
χ
[
ω
])
13
ˆ
c
in
κ
R
γ
m
(
χ
[
ω
])
14
ˆ
c
in
.
(S.15)
The transmission spectrum (driven response) is given by
T
[
ω
] =
κ
L
κ
R
(
χ
[
ω
])
11
.
(S.16)
For our system,
n
th
R
= 0
, so the symmetric noise spectral density is given by
̄
S
R
[
ω
] =
1
2
dt
〈{
ˆ
d
R,
out
[0]
,
ˆ
d
R,
out
[
t
]
}〉
e
iωt
=
1
2
+
S
R
[
ω
]
,
(S.17)
where
S
R
[
ω
] =
1
2
dt
ˆ
d
R,
out
[0]
ˆ
d
R,
out
[
t
]
e
iωt
=
κ
R
κ
|
(
χ
[
ω
])
11
|
2
n
th
c
+
κ
R
κ
|
(
χ
[
ω
])
12
|
2
(
n
th
c
+ 1
)
+
κ
R
γ
m
|
(
χ
[
ω
])
13
|
2
n
th
m
+
κ
R
γ
m
|
(
χ
[
ω
])
14
|
2
(
n
th
m
+ 1
)
,
(S.18)
For
δ
= 0
, the expressions can be simplified to
T
[
ω
] =
2
κ
L
κ
R
(
γ
m
2
)
4
G
2
+ [
κ
2
i
(
ω
+ ∆)] (
γ
m
2
)
,
(S.19)
S
R
(
ω
) =
κ
R
4
γ
m
[
γ
m
κn
th
c
+ 4
G
2
n
th
m
+ 4
G
2
+
(
n
th
m
+ 1
)]
+ 16
κn
th
c
ω
2
|
4
G
2
+ (
κ
+ 2
) [
γ
m
+ 2
i
(
ω
+ ∆)]
|
2
,
(S.20)
where
G
2
=
G
2
G
2
+
. For both
δ
= 0
and
∆ = 0
, the mechanical quadrature spectra are
̄
S
X
1
,
2
[
ω
] =
1
2
dt
〈{
ˆ
X
1
,
2
(
t
)
,
ˆ
X
1
,
2
(0)
}〉
e
iωt
= 4
x
2
zp
4
κ
(
G
G
+
)
2
(
n
th
c
+
1
2
)
+
γ
m
(
κ
2
+ 4
ω
2
)
(
n
th
m
+
1
2
)
[4
G
2
+
γ
m
κ
]
2
+ 4 (
γ
2
m
+
κ
2
8
G
2
)
ω
2
+ 16
ω
4
.
(S.21)
The mechanical quadrature fluctuations are obtained by integrating the mechanical quadrature
spectra
ˆ
X
2
1
,
2
=
2
π
̄
S
X
1
,
2
(
ω
)
=
x
2
zp
4 (
G
G
+
)
2
κ
(
2
n
th
c
+ 1
)
+ [4
G
2
+
κ
(
κ
+
γ
m
)]
γ
m
(
2
n
th
m
+ 1
)
(
κ
+
γ
m
) (4
G
2
+
κγ
m
)
,
(S.22)
1.3 Mechanical spring constant and resonance frequency
In this section, we confirm that the interaction with the cavity does not appreciably modify the
mechanical frequency and spring constant; this then confirms that the zero-point position vari-
ance
x
zp
=
~
ω
m
/
2
k
is also not changed. Starting from the solution to the quantum Langevin
equations in frequency space, we express the mechanical operator equation using the terms of the
scattering parameters,
ˆ
c
=
χ
31
κ
ˆ
d
in
+
χ
32
κ
ˆ
d
in
+
χ
33
γ
m
ˆ
c
in
+
χ
34
γ
m
ˆ
c
in
,
(S.23)
where all explicit frequency dependence has been omitted. In this notation, the mechanical sus-
ceptibility is given by the scattering factor
χ
33
. In Fig. S5, we plot the mechanical density of states
both with and without the optomechanical interaction.
We also explicitly calculate the change in the mechanical frequency due to the interaction with
the cavity by looking at the corresponding ”self energy” (i.e. difference in inverse mechanical
susceptibilities with and without the cavity, see e.g. (
19
)). With the system parameters taken at
optimal squeezing, the interaction with the cavity alters the resonance frequency by a factor of
10
3
.
Having confirmed that the mechanical resonance frequency is almost unaltered, we next con-
sider the change in the mechanical spring constant, i.e. the response of the mechanical position to
a DC force. We take the average of Eq. S.23, setting the average of
γ
m
ˆ
c
in
=
iFx
zp
/
~
, where F
represents a DC classical force. Solving for position at zero frequency in the lab frame,
ˆ
x
[0]
=
2
Fx
2
zp
~
{
Im
(
χ
33
[0])
Im
(
χ
34
[0])
}
.
The spring constant
k
is defined in terms of the position and force terms from above,
ˆ
x
[0]
/F
=
1
/k
. For the mechanics absent the optomechanical interaction,
ˆ
x
[0]
/F
= 1
/
(
2
m
)
as ex-
pected. Including the optomechanical interaction and taking parameters corresponding to the opti-
mal squeezing generation, we find that
ˆ
x
[0]
/F
is modified by 0.1%.
We have now shown that the both the spring constant and mechanical resonance frequency
differs by a factor of
10
3
when the optomechanical interaction is taken into account. The behavior
of the mechanics is insignificantly modified by the strong microwave pumps implying that there is
no modification of the zero-point scale.
1.4 Calculation of bad cavity effects on the amount of squeezing
In the previous sections, we have focused on the good cavity limit where
κ/ω
m

1
, such that
spurious off-resonant interactions described by
ˆ
H
CR
(cf. Eq. (S.5)) have been omitted. In this
section, we briefly comment on how to include bad cavity effects. As already mentioned, these
can become important if the sideband parameter
κ/ω
m
6
1
and possibly alter the lineshape of
the microwave noise spectrum or degrade the amount of mechanical squeezing. For our devices,
κ/ω
m
1
/
8
, so the bad cavity effects are small, but not negligible.
In the frequency domain, the explicit time-dependence of
ˆ
H
CR
couples system operators at
different frequencies to each other,
ˆ
D
CR
[
ω
] =
χ
CR
[
ω
]
·
L
CR
·
ˆ
D
CR
,
in
[
ω
]
.
(S.24)
Here,
ˆ
D
CR
[
ω
]
contains infinitely many sidebands detuned by
Ω = 2 (
ω
m
+
δ
)
,
ˆ
D
CR
[
ω
] =
(
...
ˆ
D
[
ω
2Ω]
,
ˆ
D
[
ω
Ω]
,
ˆ
D
[
ω
]
,
ˆ
D
[
ω
+ Ω]
,
ˆ
D
[
ω
+ 2Ω]
...
)
,
while
ˆ
D
[
ω
]
is defined in the same manner as in Sec. 1.2. Note that we do not report
χ
CR
or
L
CR
as these large matrices are not very enlightening. In order to solve the equations of motion, we
truncate the number of sidebands that we take into account, i.e. we truncate the length of
ˆ
D
CR
to the
n
th
sideband at frequency
(
ω
m
±
n
Ω)
. As the analytic solutions are unwieldy even for first
order corrections, we instead numerically calculate the spectrum at frequencies specified by the
data.
Similarly, the mechanical quadrature spectra
̄
S
X
1
,
2
can be calculated including bad cavity ef-
fects. Again, we numerically derive the spectra then extract
X
2
1
,
2
=
2
π
̄
S
X
1
,
2
via numerical
integration over a span 2000 times the total mechanical linewidth.
2 Device fabrication
We start with a 525
μ
m thick
<
100
>
-oriented high resistivity (
>
10 k
·
cm) silicon wafer. Af-
ter initial surface preparation, a 100 nm layer of aluminum is DC magnetron sputtered in a UHV
chamber with base pressure of
10
9
Torr. The bottom layer is patterned via contact photolithog-
raphy followed by two-step wet etching in Transene Al Etchant A and MF-319. Next, we spin and
pattern S1813 which acts as a sacrificial layer in the capacitor gap and a protection layer for the rest
of the bottom layer pattern. In order to thin down the sacrificial layer, we flood expose the S1813
prior to development. Before processing the top aluminum layer, we use a short O
2
plasma etch
to increase adhesion between the sacrificial and top layer. A 100 nm aluminum layer is sputtered
and patterned under the same procedure for the bottom layer. The resulting device is cleaned and
released in an overnight soak in Remover-PG followed by critical point drying and a final short O
2
plasma clean.
3 Measurements
3.1 Initial calibration measurements
We begin by performing two measurements that calibrate the pump powers detected at the output
of our measurement chain,
P
±
=
gain
×
~
ω
±
×
κ
R
±
n
±
p
, in terms of enhanced optomechanical
coupling rates,
G
±
, as well as the effective intracavity photon levels,
±
n
±
p
. Here,
±
is a correc-
tion factor that modifies the cavity transmission off resonance (
29
) and has no significance in the
following analysis.
With scanning homodyne detection (i.e., via a driven response), we first measure the mechani-
cal linewidth,
γ
tot
=
γ
m
+
γ
opt
, as we increase the power,
P
, of a single pump red-detuned from
the cavity center by
ω
m
. Here,
γ
opt
= 4
G
2
is the optically-induced mechanical damping. When
γ
tot
is much less than
κ
, the mechanical response is a simple Lorentzian dip (Fig. S2A). In this
regime, we fit
γ
opt
vs.
P
±
to obtain a calibration for
G
2
, as seen in Fig. 2B. When
γ
tot
becomes
comparable to
κ
, the cavity experiences mode-splitting, and a Lorentzian model can no longer de-
scribe the mechanical response (Fig. S2B). We thus fit Eq. S.19 to the transmission, with
G
+
= 0
and
δ
= 0
. The values of
G
extracted from the fit to the full model is in good agreement with the
linear fit to
γ
opt
at lower powers, as seen in Fig. S3.
Next, we place two balanced pumps, detuned from cavity center by
±
(
ω
m
+ 2
π
×
500 Hz)
and
with powers
P
±
, at sufficiently low powers so as not to add any damping or amplification of the
thermal noise, and we measure the integrated mechanical noise power of up- and down-converted
motional sidebands,
P
±
m
, over a range of cryostat temperatures
T
. Due to weak temperature and
power dependence of
κ
(
21
), we monitor the cavity linewidth at each measurement power or tem-
perature. The results of these calibrations are cast in a linear form and fit with ordinary least
squares to extract the calibration factors
a
,
b
and
b
+
,
(
κ
̄
κ
)
γ
opt
=
4
G
2
̄
κ
=
a
(
4
̄
κ
)
P
,
(S.25)
(
κ
̄
κ
)
2
P
m
P
±
=
b
±
(
2
̄
κ
)
2
k
B
T
~
ω
m
,
(S.26)
where
̄
κ
is the cavity linewidth averaged over the respective parameter range. We find
a
= (3
.
25
±
0
.
09)
×
10
16
rad
2
s
2
W
1
,
b
= (3
.
82
±
0
.
14)
×
10
4
rad
2
s
2
,
b
+
= (6
.
84
±
0
.
22)
×
10
4
rad
2
s
2
.
We now are able to formulate the pump-dependent model parameters in terms of
P
±
,
G
2
=
a
×
P
,
(S.27)
G
2
+
=
a
(
b
b
+
)
×
P
+
,
(S.28)
n
p
=
(
a
b
)
×
P
.
(S.29)
Eq. (S.28) follows from the balancing condition
(
P
P
+
)
balanced
=
(
b
b
+
)
.
3.2 Noise spectrum measurement
We first measure the complex transmission through the system at each reported power ratio. Setting
the enhanced optomechanical coupling rates via Eqs. (S.27-S.29), we fit the transmission spectrum
to Eq. (S.19) via nonlinear least squares estimation and extract the frequency of the microwave
resonator
ω
c
, the cavity linewidth
κ
, the frequency of the mechanical oscillator
ω
m
, and the pump
detunings
,
δ
. The two pump tones are iteratively aligned to overlap the mechanical sidebands at
the center of the cavity to ensure that
δ
is close to 0 (Fig. S4).
We then measure the microwave noise spectrum via linear detection, keeping the same two-tone
pump configuration as above. The measured spectrum is given by
̄
S
out
(
ω
) =
S
0
(
ω
) +
S
R
(
ω
)
,
(S.30)
where
S
R
(
ω
)
is the noise spectrum of the electro-mechanical system and
S
0
(
ω
)
is the noise floor
of the system. The noise floor is dominated by the noise figure of the cryogenic HEMT amplifier
in addition to smaller power-dependent offsets due to phase noise from the entire amplifier chain.
We spend an equal time interleaving measurements of the pumped and unpumped noise spectra
over the same bandwidth. We subtract off the unpumped floor, then account for power-dependent
amplifier effects by removing a linear floor offset that we fit over a span
7
times greater than the
cavity linewidth. The linear offset matches independent measurements of the phase noise from our
room temperature amplifier with matching pump configuration.
4 Error analysis and fitting of the output spectrum
Essential to any claim of sub-zero-point squeezing is the error bar for the reported quadrature
occupation. Here, we consider a systematic approach to incorporate the uncertainty from all the
sources of our measurement, including systematic calibration error, measurement noise, and the
uncertainty from fitting the model to a measured noise spectrum. This problem has been addressed
by Bayesian analysis techniques that explicitly incorporate all known sources of error. In the
following, we largely follow the analysis outlined in Ch. 3 of (
30
). Our purpose for using this
analysis is to address the issue of estimating error bars from nonlinear fitting with a fit model that
also has uncertainty.
In what follows, we develop statistical estimators for the quadrature occupations,
X
2
1
,
2
, from
two sets of measurements: the detected noise spectrum and the system calibrations. Here, sys-
tem calibration refers to the combination of initial calibrations
(
a,b
,b
+
)
, driven response data
(
κ,
)
and power detection
(
P
,P
+
)
. We refer to such parameters as
β
=
{
a,b
,b
+
,κ,
,δ,P
,P
+
}
.
The only remaining unknowns are the bath contributions, here denoted as
α
=
{
n
th
c
m
n
th
m
}
.
To systematically incorporate the uncertainty from our calibrations and spectrum measure-
ments, we consider the Bayesian posterior distribution
p
(
α,β
|
D,I
) =
1
Z
p
(
D
|
α,β,I
)
p
(
α,β
)
,
(S.31)
where
D
is the observed noise data,
I
is the set of all assumptions required for this analysis,
Z
=
p
(
D
)
is a normalization constant that is not necessary for sampling of the posterior,
p
(
D
|
α,β,I
)
is the likelihood function, and
p
(
α,β
)
is the prior distribution for
α
,
β
.
The prior distribution captures how well we have confined our calibrations in parameter space.
Assuming all system parameters are independent, the prior simplifies to a product of single-
parameter normal distributions, i.e.
p
(
β
)
is a product of Gaussian distributions with mean and
variance set by the statistical estimators for each system calibrations. For the unknowns,
p
(
α
)
is
the product of uninformed Jeffreys priors; these priors are uniform in log space and here set to
span a decade above and below initial estimates for
n
th
c
,
γ
m
n
th
m
.
The likelihood captures how well the data matches the noise spectrum model with specified
α
,
β
. We calculate data residuals by subtracting the full noise spectrum model, including bad-cavity
effects as discussed in Sec. 1.4, from the detected noise. Next, we assume the measurement noise is
independent and Gaussian distributed with constant variance. Hence, the likelihood is the product
of residual probabilities derived from
N
(0
)
, where
σ
is sampled from noise data over a 150 kHz
window detuned outside cavity center by
±
3
κ
.
Since the posterior distribution is difficult to calculate analytically, we instead model the pos-
terior via an affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampler (
31
). We
implement this calculation with
emcee
, an open-source Python package developed in the astron-
omy community with over 300 citations since 2012 (
32
). With
emcee
, we generate a sufficiently
large number of of pseudo-random parameter chains
(
α
i
i
)
sampled from the posterior distribu-
tion. For the calculation, we initialize
10
2
walkers and run for a minimum of
10
3
steps. We discard
the first half to ensure that the resulting distributions are steady state (allowing initial transients to
relax) but maintain large enough sample size to render the Monte Carlo uncertainty negligible.
Finally, we calculate expectation values and 1-
σ
intervals for
n
th
c
,
γ
m
n
th
m
and
X
2
1
,
2
. For
n
th
c
and
γ
m
n
th
m
, we construct the marginalized distributions from their respective Markov chains and
then tabulate the statistical estimators for mean and variance. For the mechanical quadratures, we
calculate expectation values for functions of system parameters,
f
(
α,β
)
, with function evaluation
over the entire MCMC ensemble,
f
=
f
(
α,β
)
p
(
α,β
|
D,I
)
dαdβ,
(S.32)
'
1
N
N
i
=1
f
(
α
i
i
)
.
(S.33)
The mean and standard deviation for
X
2
1
,
2
are generated via Eq. (S.33) with
f
(
α,β
)
set to the
mechanical quadrature functions discussed in Sec. 1.4.
ω
c
ω
m
ω
m
δ
Δ
δ
ω
-
ω
+
Fig. S1: Definition of detuning parameters for an optomechanical system with a two-tone drive.
f
!
f
c
(
H
z
)
-1000
-500
0
500
1000
j
S
2
1
j
0.01
0.02
0.03
0.04
0.05
1.2e+03
4.7e+03
1.2e+04
1.9e+04
3.0e+04
4.8e+04
f
!
f
c
(
H
z
)
#
10
5
-3
-2
-1
0
1
2
3
0.01
0.02
0.03
0.04
0.05
6.5e+05
3.9e+06
6.1e+06
1.0e+07
1.6e+07
2.5e+07
B
A
Fig. S2: Cavity transmission spectra in the presence of a microwave drive at frequency
ω
=
ω
c
ω
m
. A. Transmission spectra in weak-driving regime, with
n
p
ranging from 1.2e3 to 4.8e4.
In this regime, the spectrum can be described by a Lorentzian dip with an effective mechanical
linewidth
γ
tot
=
γ
m
+
γ
opt
. Increasing the pump photon number increases the optical damping rate
and broadens the mechanical signal. B. Transmission spectra in the strong-driving regime, with
n
p
ranging from 6.5e5 to 2.5e7. When the effective mechanical linewidth
γ
tot
is comparable to the
cavity linewidth
κ
, the resonance starts to split into normal modes, which are the hybrids of the
mechanical oscillations and the microwave resonance. In this regime, the full model (S.19) is used
to describe the transmission spectrum.
P
thru
(W)
10
-10
10
-8
10
-6
10
-4
(G
2
-G
fit
2
)/
"
G
2
-1
-0.5
0
0.5
1
n
p
10
2
10
4
10
6
Fig. S3: Normalized residuals of the enhanced electromechanical coupling rate. Purple circles:
weak-driving regime in which the mechanical susceptibility is a simple lorentzian. Red cir-
cles: strong-driving regime in which the mechanical linewidth becomes comparable to the cavity
linewidth and must be extracted using a full model. The linearity of our device at high pump pow-
ers is shown by calculating the normalized residuals of these strong-driving points to the fit line
from the weak-driving regime.
/
/
.
tot
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
min|S
21
| (dB)
-65
-60
-55
-50
-45
-40
f-f
c
(kHz)
-15,
-10,
-5
0
5
10
15
|S
21
| (dB)
-70
-60
-50
-40
-30
-20
-2000
-1000
-500
0
500
1000
2000
B
A
/
(Hz)
Fig. S4: Sideband alignment precision. (A) Transmission spectra in the presence of two drives
at frequencies equal to
ω
±
; the dip in the spectrum is due to the two tones’ electromechanically
induced transparency effect. The detuning
δ
is given by
(
ω
+
ω
2
ω
m
)
/
2
. The transmission
at
ω
c
is minimized when the up-converted mechanical sideband from the red pump and the down-
converted mechanical sideband from the blue pump are perfectly overlapped. (B) The minimum
value of the transmission spectrum in (A) vs. detuning. The precision of the alignment is within a
few percent of the total mechanical linewidth
γ
tot
=
γ
m
+
4
κ
(
G
2
G
2
+
)
.
0.8
0.9
1.0
1.1
1.2
ω/ω
m
10
-12
10
-11
10
-10
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
Re(
χ
33
)
bare
OM
Fig. S5: The mechanical density of states, defined as Re
(
χ
33
)
, both with (“OM”) and without
(“bare”) the optomechanical interaction. At optimal squeezing parameters, the mechanics is sig-
nificantly damped by the optomechanical interaction, however the location of the mechanical peak,
and hence the mechanical resonance frequency, is insignificantly modified by the interaction.