Supplementary Information for “Quantum nondemolition measurement of mechanical
squeezed state beyond the 3 dB limit”
C. U. Lei,
1
A. J. Weinstein,
1
J. Suh,
2
E. E. Wollman,
1
A. Kronwald,
3
,
4
F. Marquardt,
3
,
4
A. A. Clerk,
5
K. C. Schwab
1
1
Applied Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Korea Research Institute of Standards and Science, Daejeon 305-340, Republic of Korea
3
Friedrich-Alexander-Universit ̈at Erlangen-N ̈urnberg, Staudtstr. 7, D-91058 Erlangen, Germany
4
Max Planck Institute for the Science of Light G ̈unther-Scharowsky-Straße 1/Bau 24, D-91058 Erlangen, Germany and
5
Department of Physics, McGill University, Montreal, Quebec, H3A 2T8 Canada
(Dated: July 26, 2016)
I. THEORY
A. Ideal two tones optomechanical Hamiltonian
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
,
(1)
where ˆ
a
(
ˆ
a
†
)
is the annihilation (creation) operator of the intra-cavity field,
ˆ
b
(
ˆ
b
†
)
is the mechanical phonon annihila-
tion (creation) operator, and
g
0
is the bare optomechanical coupling between the cavity and the mechanical oscillator.
ˆ
H
drive
describes the external driving.
The device studied in this work is a two ports optomechanical system. Microwave tones are applied from the
left port, which we designate (
L
). In this section, we consider a system driven by two microwave tones. The drive
Hamiltonian reads
ˆ
H
drive
=
~
√
κ
L
∑
ν
=
±
α
ν
(
ˆ
ae
iω
ν
t
+ ˆ
a
†
e
−
iω
ν
t
)
,
(2)
where
ω
±
=
ω
c
+ ∆
±
(
ω
m
+
δ
) and
α
±
are the blue and red pump amplitudes at the input port. In the following,
we apply the 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
b
. 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
=
ˆ
H
RWA
+
ˆ
H
CR
.
(3)
Here,
ˆ
H
RWA
=
−
~
∆
ˆ
d
†
ˆ
d
−
~
δ
ˆ
b
†
ˆ
b
−
~
[(
G
+
ˆ
b
†
+
G
−
ˆ
b
)
ˆ
d
†
+
(
G
+
ˆ
b
+
G
−
ˆ
b
†
)
ˆ
d
]
(4)
describes the resonant part of the linearized optomechanical interaction whereas
ˆ
H
CR
=
−
~
[
G
+
e
−
2
i
(
ω
m
+
δ
)
t
ˆ
b
+
G
−
e
2
i
(
ω
m
+
δ
)
t
ˆ
b
†
]
ˆ
d
†
−
~
[
G
+
e
2
i
(
ω
m
+
δ
)
t
ˆ
b
†
+
G
−
e
−
2
i
(
ω
m
+
δ
)
t
ˆ
b
]
ˆ
d
(5)
describes off-resonant optomechanical interactions. Note that
G
±
=
g
0
̄
a
±
describes the driven-enhanced optome-
chanical 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. In the following analysis, we consider the good cav-
ity limit (
ω
m
κ
). At this limit, the off-resonant part of the Hamiltonian can be neglected by the rotating wave
approximation (RWA).
B. Mechanical parametric modulation
In addition to the ideal optomechanical interaction, mechanical parametric modulation is observed in the experi-
ment. This spurious mechanical parametric effect can be induced by thermal effects or nonlinearities [1, 2]. To take
this effect into account, we phenomenologically include the mechanical parametric interaction
ˆ
H
para
=
−
~
λ
(
e
iψ
ˆ
b
2
+
e
−
iψ
ˆ
b
†
2
)
,
(6)
where
λ
is the amplitude of the parametric interaction,
ψ
is the relative phase between the parametric drive and the
squeezing pump.
2
C. Quantum Langevin equations
The linearized quantum Langevin equations read
̇
ˆ
d
=
−
(
κ
2
−
i
∆
)
ˆ
d
+
i
(
G
−
ˆ
b
+
G
+
ˆ
b
†
)
+
√
κ
ˆ
d
in
,
(7)
̇
ˆ
b
=
−
(
Γ
m
2
−
iδ
)
ˆ
b
−
i
2
λe
−
iψ
ˆ
b
†
+
i
(
G
−
ˆ
d
+
G
+
ˆ
d
†
)
+
√
Γ
m
ˆ
b
in
.
(8)
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
′
)
,
(9)
[
ˆ
b
in
(
t
)
,
ˆ
b
†
in
(
t
′
)
]
=
δ
(
t
−
t
′
)
,
(10)
〈
ˆ
d
†
σ
′
,
in
(
t
)
ˆ
d
σ,
in
(
t
′
)
〉
=
n
th
σ
δ
σ,σ
′
δ
(
t
−
t
′
)
,
(11)
〈
ˆ
b
†
in
(
t
)
ˆ
b
in
(
t
′
)
〉
=
n
th
m
δ
(
t
−
t
′
)
,
(12)
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
.
D. Optomechanical output spectrum and mechanical spectrum
In this section, we derive the optomechanical output spectrum and the mechanical quadrature spectrum. For
this, we solve the quantum Langevin equations (Eqs. 7, 8) in Fourier space. It is convenient to define the vectors
D
=
(
ˆ
d,
ˆ
d
†
,
ˆ
b,
ˆ
b
†
)
T
,
D
in
=
(
ˆ
d
in
,
ˆ
d
†
in
,
ˆ
b
in
,
ˆ
b
†
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
]
,
(13)
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
)
i
2
λe
−
iψ
iG
+
iG
−
−
i
2
λe
iψ
Γ
m
2
−
i
(
ω
−
δ
)
−
1
.
(14)
In the experiment, we measure the output microwave spectrum through the undriven (right) cavity port. One finds
the output 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
−
√
κ
R
Γ
m
(
χ
[
ω
])
13
ˆ
b
in
−
√
κ
R
Γ
m
(
χ
[
ω
])
14
ˆ
b
†
in
.
(15)
The transmission spectrum (driven response) is
S
21
[
ω
] =
−
√
κ
L
κ
R
(
χ
[
ω
])
11
.
(16)
The symmetric noise spectral density is
̄
S
R
[
ω
] =
1
2
∫
dt
〈{
ˆ
d
†
R,
out
[0]
,
ˆ
d
R,
out
[
t
]
}〉
e
iωt
=
1
2
+
κ
R
S
[
ω
]
,
(17)
where
S
[
ω
] =
κ
|
(
χ
[
ω
])
11
|
2
n
th
c
+
κ
|
(
χ
[
ω
])
12
|
2
(
n
th
c
+ 1
)
+ Γ
m
|
(
χ
[
ω
])
13
|
2
n
th
m
+ Γ
m
|
(
χ
[
ω
])
14
|
2
(
n
th
m
+ 1
)
.
(18)
3
The mechanical quadrature spectrum is
̄
S
X
φ
[
ω
] =
1
2
∫
dt
〈{
ˆ
X
φ
(
t
)
,
ˆ
X
φ
(0)
}〉
e
iωt
,
(19)
where
ˆ
X
φ
=
x
zp
(
ˆ
be
iφ
+
ˆ
b
†
e
−
iφ
)
. The quadrature variance is given by the integral
〈
ˆ
X
2
φ
〉
=
∫
dω
2
π
̄
S
X
φ
(
ω
)
.
(20)
In some pump configurations, we can simplify the results. For
δ
= 0, the expressions can be simplified to
S
21
[
ω
] =
−
2
√
κ
L
κ
R
(Γ
m
−
2
iω
)
4
G
2
+ [
κ
−
2
i
(
ω
+ ∆)] (Γ
m
−
2
iω
)
,
(21)
S
[
ω
] =
4Γ
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
iω
) [Γ
m
+ 2
i
(
ω
+ ∆)]
|
2
,
(22)
where
G
2
=
G
2
−
−
G
2
+
. For both
δ
= 0 and ∆ = 0, the mechanical quadrature spectra and the quadrature variances
are
̄
S
X
1
,
2
[
ω
] = 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
.
(23)
〈
ˆ
X
2
1
,
2
〉
=
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
+
κ
Γ
m
)
,
(24)
in the regime
κ
G,
Γ
m
, Eq. (24) reduced to Eq. (4) in the main text.
II. MEASUREMENT CIRCUIT
The schematic of the measurement circuit is shown in Fig. S1 We cool the device down to 10 mK with a dilution
refrigerator. In the experiment, up to four microwave drives are applied to the device. Because the excess phase
noise from the microwave sources at the cavity resonance can excite the cavity and degrade the squeezing, a tunable
notch filter cavity is used to provide more than 50 dB noise rejection at the cavity resonance frequency
ω
c
. The input
microwave pumps are then attenuated by about 40 dB at different temperature stages in the cryostat to dissipate the
Johnson noise from higher temperature, keeping the input microwave noise at the shot noise level. The output signal
passes through two cryocirculators at 50mK, then amplified by a cryogenic high electron-mobility transistor amplifier
(HEMT) at 4.2 K and a low noise amplifier at room temperature for analysis. During the measurement of the noise
spectrum, we continuously monitor the phase difference between the squeezing pumps and the BAE probes. The
beat tones of the pumps and the probes are acquired by microwave detection diodes, then fed into the sub-harmonic
circuits to halve the frequencies. The relative phase between the resulting beat tones are compared and measured by
the lock-in. A computer is used to generate the error signal and feedback to the sources to keep the phase drift within
half degrees.
III. CALIBRATIONS
A. Calibrations of the squeezing output spectrum
In the experiment, we spend an equal time interleaving measurement to measure the pumped noise spectrum
̄
S
meas
[
ω
] and the unpumped noise spectrum
̄
S
0
[
ω
] at the output of the measurement chain. The unpumped noise
spectrum
̄
S
0
[
ω
] is the noise floor of the system which is dominated by the noise figure of the cryogenic HEMT
amplifier. The difference of the pumped and unpumped noise spectra is related to the output noise spectrum of the
optomechanical system by
∆
̄
S
meas
[
ω
] =
̄
S
meas
[
ω
]
−
̄
S
0
[
ω
] =
G
[
ω
c
]
κ
R
~
ω
c
S
[
G
−
,G
+
,
∆
,δ,κ,
Γ
m
,n
th
c
,n
th
m
,ω
]
,
(25)
4
Pump
cryostat
device
spectrum
analyzer
network
analyzer
4 - 20 mK
10 mK
50 mK
4 K
lter cavity
Probe
φ
φ
Lock-in
ω
/
2
ω
/
2
FIG. S1. Schematic of the measurement circuit. See text for detail.
0.5
1
1.5
2
2.5
3
3.5
G
2
-
, G
2
+
(rad/s)
2
x
10
9
0
2
4
6
8
10
f-f
c
(kHz)
-60
-30
0
30
60
|S
21
|
0.05
0.1
f-f
c
(kHz)
-2
-1
0
1
2
|S
21
|
0
0.05
∆
=
0
2
δ
ω
c
−
ω
m
−
δ
ω
c
+
ω
m
+
δ
ω
c
−
ω
m
(a)
(b)
(c)
T (mK)
10
100
10
-5
10
-4
f-f
m
(Hz)
-80
-40
0
40
80
S (aW/Hz)
-1
0
1
2
∆
=
0
(d)
, P
+
(nW)
P
-
P
m
P
(
κ
2
+
4∆
2
̄
κ
2
)
FIG. S2. Calibrations of the mechanical squeezing experiment. (a) Pump configuration of the enhanced optomechanical
coupling (
G
−
) calibration. (b) Pump configuration of the enhanced optomechanical coupling (
G
+
) calibration. (c) Calibrations
of the enhanced optomechanical couplings
G
±
, the inserts are the transmission spectrums corresponding to the solid circles.
(d) Calibration of the normalized motional sideband power, the insert is the sideband spectrum at the base temperature.
where
G
[
ω
] is the gain of the measurement chain which is flat in the measurement bandwidth of the experiment
and
κ
R
is the coupling rate to the output port of the device. In order to fit the measured noise spectrum with the
optomechanical model (Eq. (18)), an independent measurement of the prefactor
G
[
ω
c
]
κ
R
~
ω
c
is required. Which can
be obtained from the calibrations of the enhanced optomechanical couplings and the thermal calibration.
1. Calibrations of the enhanced optomechanical couplings
To calibrate the enhanced optomechanical couplings
G
±
, we measure the transmitted power of the drive tones at
the output of the measurement chain
P
±
, which is related to the intracavity pump photon number by
P
±
=
G
[
ω
±
]
κ
R
~
ω
±
λ
[
ω
±
]
n
±
p
,
(26)
where
λ
[
ω
±
] are the correction factors due to the parasitic channel [3]. The square of the enhanced optomechanical
couplings are related linearly to the transmitted pump powers by
G
2
±
=
g
2
0
n
±
p
=
a
±
×
P
±
,
(27)
where the
a
±
=
1
G
[
ω
±
]
κ
R
~
ω
±
g
2
0
λ
[
ω
±
]
is the calibrations of the enhanced optomechanical couplings.
We start with the calibration of the enhanced optomechanical coupling
G
−
induced by the red detuned tone. To
do that, a single red detuned tone is applied at
ω
c
−
ω
m
with transmitted power
P
−
(Fig. S2a). Then, a network
analyzer is used to generate a weak probe and sweep it through the center of the cavity resonance to measure the
transmission spectrum of the mechanical sideband. The enhanced optomechanical coupling rate
G
−
can be extracted
by fitting the transmission spectrum with the optomechanical model (16). By measuring the transmission spectrum
5
with various transmitted power
P
−
and fitting with the linear relation (27) (the red line in Fig. S2c), we obtain the
calibration
a
−
= (7
.
49
±
0
.
10)
×
10
17
rad
2
s
−
1
W
−
1
and the intrinsic mechanical linewidth Γ
m
= 2
π
×
8 Hz.
A similar method is used to calibrate the enhanced optomechanical coupling
G
+
induced by the blue detuned tone.
In this case, a blue detuned tone is placed at
ω
c
+
ω
m
+
δ
with transmitted power
P
+
and
δ
= 2
π
×
30kHz
κ
. Because
the blue detuned tone would amplify the mechanical motion and narrow the mechanical linewidth, the mechanical
resonator becomes unstable when the cooperativity
C
+
=
4
G
2
+
κ
Γ
m
approaches to unity. In order to keep the mechanics
stable, a constant red detuned tone is applied at
ω
c
−
ω
m
−
δ
to damp the mechanical motion (Fig. S2b). Similar to
the calibration of
G
−
, we measure the transmission spectrum of the mechanical sidebands with the spectrum analyzer
and extract the enhanced optomechanical coupling rate
G
+
from the fit at various transmitted power
P
+
. The linear
fit (blue line in Fig. S2c) gives
a
+
= (3
.
23
±
0
.
07)
×
10
18
rad
2
s
−
1
W
−
1
.
2. Thermal calibration
Having calibrated the enhanced optomechanical couplings, we turn to the thermal calibration of the motional
sideband noise power. To do that, a single red detuned tone is placed at
ω
−
=
ω
c
−
ω
m
with sufficiently small pump
power
P
−
such that the optomechanical damping effect is negligible (Γ
−
opt
=
4
G
2
−
κ
Γ
m
). We then measure the noise
power of the up-converted motional sideband
P
−
m
over a range of calibrated cryostat temperature
T
(Fig. S2d). Due
to the weak temperature dependence of the cavity linewidth
κ
, we monitor the cavity linewidth at each measurement
temperature. The resulting normalized sideband power is given by
(
4∆
2
+
κ
2
̄
κ
2
)
P
m
P
−
=
b
−
(
2
̄
κ
)
2
k
B
T
~
ω
m
,
(28)
where ∆ =
ω
−
−
ω
c
+
ω
m
which is equal to zero in this case, ̄
κ
is the average value of the cavity linewidth over
the respective temperatures and
b
−
=
G
[
ω
c
]
ω
c
G
[
ω
−
]
ω
−
g
2
0
λ
[
ω
−
]
is the thermal calibration. The linear fit in Fig. S2d gives
b
−
= (2
.
53
±
0
.
07)
×
10
5
(rad/s)
2
, which enables us to convert the normalized noise power into quanta. The prefactor
G
[
ω
c
]
κ
R
~
ω
c
is given by the ratio of the thermal calibration
b
−
and the calibration of the enhanced optomechanical
coupling
a
−
(i.e.
G
[
ω
c
]
κ
R
~
ω
c
=
b
−
/a
−
), which allow us to relate the measured noise spectrum to the optomechanical
model
∆
̄
S
meas
[
ω
] =
b
−
a
−
̄
S
[
G
−
,G
+
,
∆
,δ,κ,
Γ
m
,n
th
c
,n
th
m
,ω
]
.
(29)
3. Fitting procedure
In order to extract the cavity occupation
n
th
c
and the phonon bath occupation
n
th
m
from the output spectrum. We
need to determine the enhanced optomechanical couplings
G
±
, the detunings ∆ and
δ
, the cavity linewidth
κ
and
the intrinsic mechanical linewidth Γ
m
. To obtain these parameters, we measure the transmitted pump power of the
microwave drives at each pump configuration, from which we obtain the enhanced optomechanical couplings
G
±
with
the calibrations
a
±
. Since the resonance frequencies of the cavity and the mechanical mode change with the powers
of the microwave drives due to thermal effects [2] and nonlinearties [1], to precisely align the frequencies of the drive
tones at
ω
c
±
ω
m
, we measure the transmission spectrum to extract the detunings and correct the frequencies of the
drives. Fig. S3 shows examples of the measured driven responses (dots) and the corresponding fits (solid curves) at
various pump photon ratios. By fitting the transmission spectrum with Eq. (16), we can extract the detunings (
δ,
∆)
and the cavity linewidth
κ
, which enable us to correct the frequencies of the drive tones. By iterating this procedures,
we can precisely set the powers and the frequencies of the microwave drives. Together with the intrinsic mechanical
linewidth Γ
m
obtained from the calibration, we can fit the output noise spectrum with the optomechanical model
and extract the occupation factors (
n
th
c
, n
th
m
) with Eq. (29). Fig. S4 shows examples of the noise spectra and the
corresponding fits.
B. Calibrations of the backaction evasion spectrum
In our experiment, we perform an additional BAE measurement away from the cavity resonance to directly and
independently measure the mechanical quadratures. Since the detuning of the BAE sideband ∆ = 2
π
×
160kHz