Mechanically Detecting and Avoiding the Quantum Fluctuations
of a Microwave Field
J. Suh,
1, 2
A. J. Weinstein,
1, 2
C. U. Lei,
1, 2
E. E. Wollman,
1, 2
S. K. Steinke,
1, 3
P. Meystre,
3
A. A. Clerk,
4
and K. C. Schwab
1, 2,
∗
1
Applied Physics, Caltech, Pasadena, CA, 91125 USA
2
Kavli Nanoscience Institute, Caltech, Pasadena, CA 91125 USA
3
Department of Physics, University of Arizona, Tuscon, AZ 85721 USA
4
Department of Physics, McGill University, Montreal, QC, H3A 2T8 CA
During the theoretical investigation of the ultimate sensitivity of gravitational wave detectors
through the 1970’s and ’80’s, it was debated whether quantum fluctuations of the light field used
for detection, also known as photon shot noise, would ultimately produce a force noise which would
disturb the detector and limit the sensitivity. Carlton Caves famously answered this question with
“They do.” [1] With this understanding came ideas how to avoid this limitation by giving up
complete knowledge of the detector’s motion[2–4]. In these back-action evading (BAE) or quantum
non-demolition (QND) schemes, one manipulates the required quantum measurement back-action
by placing it into a component of the motion which is unobserved and dynamically isolated. Using a
superconducting, electro-mechanical device, we realize a sensitive measurement of a single motional
quadrature with imprecision below the zero-point fluctuations of motion, detect both the classical
and quantum measurement back-action, and demonstrate BAE avoiding the quantum back-action
from the microwave photons by 9 dB. Further improvements of these techniques are expected to
provide a practical route to manipulate and prepare a squeezed state of motion with mechanical
fluctuations below the quantum zero-point level, which is of interest both fundamentally[5] and for
the detection of very weak forces[6].
Since the discovery of Shor’s Algorithm[7] almost 20
years ago, a major theme in physics has been about the
untapped power and benefits of quantum phenomena,
largely stemming from the resource of quantum entan-
glement. However much earlier, it was understood how
quantum physics places limits on our knowledge[8, 9].
This limitation can be useful, as in the case of quantum
cryptography schemes where the required quantum mea-
surement back-action of an eavesdropper leaves its trace
on the transmitted information, providing proof of their
snooping. For measurements of position, this limitation,
called the Standard Quantum Limit (SQL)[9] is not ben-
eficial: back-action due to the quantum nature of the
measurement field, ultimately obscures our vision for a
sufficiently sensitive measurement.
Quantum limitations on the detection of position are
no longer academic issues; in recent years, the detection
of motion has now advanced to the point that quan-
tum back-action engineering is now required to improve
the sensitivity. Detections of motion have been realized
with imprecision below that at SQL[10, 11]. Back-action
forces from the quantum noise of the detection field have
been demonstrated to drive the motion of mechanical
oscillators, first with electrons in an electro-mechanical
structure[12] and then with photons in opto-mechanical
systems [13, 14]. In this work, we demonstrate the back-
action forces due to the shot noise of microwave photons,
which are 10
4
times lower in energy than optical photons.
Strategies to manipulate the quantum measurement
back-action have included modifying the quantum fluctu-
ations of the measurement field[15, 16], and modulating
the coupling between the detection field and mechani-
cal element. The modulation of coupling can be imple-
mented by either sudden stroboscopic measurement[17,
18] or continuous two-tone BAE measurement[2–4, 6],
which we pursue here.
The system we study is a parametric transducer (Fig.
1a): a mechanical resonator (
ω
m
= 2
π
·
4.0 MHz) modu-
lates the capacitance of a superconducting electrical res-
onator (
ω
c
= 2
π
·
5.4 GHz), and modifies
ω
c
by 14 Hz
(=
g
0
/
2
π
) per
x
zp
, where
x
zp
=
√
̄
h/
(2
mω
m
)
≈
1
.
8 fm is
the amplitude of zero-point fluctuations of the mechan-
ical resonator with mass
m
. The damping rate of the
electrical resonator is
κ
= 2
π
·
0.86 MHz, which places
this system into the side-band resolved limit (
ω
m
> κ
)
required to realize BAE dynamics[6, 19]. When pumping
the transducer with microwave photons at
ω
p
=
ω
c
−
ω
m
,
the electro-mechanical coupling together with mechani-
cal motion results in frequency up-conversion of pump
photons to
ω
c
in a Raman-like process at a rate
n
m
Γ
opt
,
where
n
m
is the occupation factor of the mechanical
mode, and Γ
opt
= 4
g
2
0
n
p
/κ
with
n
p
as the number of
pump photons stored in the electrical resonator. Simi-
larly, when pumping at
ω
p
=
ω
c
+
ω
m
, photons are down-
converted at a rate (
n
m
+1)Γ
opt
. These sidebands are the
signals we analyze in this work: we use thermal motion
of the mechanical resonator at calibrated temperatures
to measure the transduction gain between the sideband
power and
n
m
[20]. Based upon the thermal calibration,
we observe a motional sideband corresponding to 7.2
±
0.2
mK at the base temperature of our refrigerator (Fig.1c,
red cross and inset). The thermal calibration also deter-
arXiv:1312.4084v1 [quant-ph] 14 Dec 2013
2
FIG. 1. Device, measurement scheme and sample character-
ization.
a
, Device electron micrograph. A parallel plate ca-
pacitor is connected to a spiral inductor, forming a lumped-
element microwave resonator. The top plate of the capaci-
tor is a compliant membrane, and we study its fundamental
mechanical resonance here. Color indicates different materi-
als used: blue - silicon, gray - aluminum.
b
, Measurement
scheme. Shot-noise-limited microwave tones are generated
with room-temperature filtering and cryogenic attenuation,
and applied to the device at 20 mK. The output microwave
field from the device is amplified by a low-noise amplifier at
4.2K, and its spectra are analyzed (Supplementary Informa-
tion).
c
, Calibration of motional sidebands against thermal
motion (blue circles and line). Inset, motional sideband at
base temperature corresponding to 7
.
2
±
0
.
2 mK.
d
, Back-
action damping and calibration for the number of pump pho-
tons. In addition to a red-detuned pump, a weak probe sweep-
ing near the electrical resonance is applied and its absorp-
tion is monitored, showing the resonant mechanical response.
Blue circle, mechanical damping rate. Blue line, back-action
damping theory fit[21]. Inset, examples of absorption spectra
at
n
p
≈
5
×
10
3
,
3
×
10
4
,
1
×
10
5
from top to bottom.
mines
g
0
, and we combine it with back-action damping
measurement[21, 22], to generate the calibration of
n
p
vs
measured microwave power (Fig.1d).
Two-tone BAE is accomplished in this system with
the application of a modulated electric field:
E
(
t
) =
E
p
cos
ω
m
t
·
cos
ω
c
t
= (
E
p
/
2) [cos(
ω
c
−
ω
m
)
t
+ cos(
ω
c
+
ω
m
)
t
].
This technique has the effect of coupling to
a single mechanical quadrature
X
1
, where
x
(
t
) =
X
1
(
t
) cos
ω
m
t
+
X
2
(
t
) sin
ω
m
t
. The BAE nature can be
understood by noting that the back-action force pro-
duced at the mechanical frequency, by the beating be-
tween voltage noise at the microwave resonance and the
pump field, produces displacements exclusively in the
X
2
quadrature[19]. In this way, one gains information about
X
1
and places the required quantum measurement back-
action into
X
2
. This fact can be also understood by
observing that
X
1
and
X
2
are constants of motion and
thus quantum non-demolition observables[3].
Recent experiments have attempted to implement two-
tone BAE demonstrating a single quadrature imprecision
close to the zero-point fluctuations, with further sensi-
tivity improvement blocked by parametric instabilities
arising from non-linearities of the coupling[20], thermal
dissipation[23], and two-level system effects[24]. In the
experiments described here, we study an improved de-
vice which largely avoids these limitations. However,
to overcome difficulties in aligning BAE tones arising
from a small mechanical frequency jitter comparable to
the mechanical damping rate of Γ
m
0
= 2
π
·
10 Hz at
20 mK, we also apply a red-detuned cooling tone at
ω
c
−
ω
m
−
35kHz to both cool the resonator from occupa-
tion factor
n
m
≈
100 to 15, and to broaden the mechan-
ical resonance from Γ
m
∼
2
π
·
10 Hz to 2
π
·
100Hz[21, 22].
Since this cooling tone is well-separated from the mea-
surement tones with respect to the mechanical linewidth
(Γ
m
35 kHz), it effectively adds mechanical damping
without perturbing our BAE measurements (Fig.3a).
To demonstrate the avoidance of measurement back-
action, we first pump the transducer with two tones of
equal power with frequencies of
ω
c
±
(
ω
m
+ ∆), where
each tone is detuned by ∆ = 5kHz from the BAE con-
figuration, producing two separate motional sidebands.
When ∆
Γ
m
, the up- and down- converted signal pho-
tons provide a measurement of both mechanical quadra-
tures, and as a result, the measurement is subject to
the usual back-action forces resulting in extra position
fluctuations,
〈
x
2
〉
ba
/x
2
zp
= 2(Γ
opt
/
Γ
m
)(2
n
c
+ 1), where
n
c
is the occupation factor of the electrical resonator,
and +1 is due to the quantum fluctuations of the mi-
crowave field. The observed motional sidebands exhibit
slight asymmetry mainly due to the interference between
microwave noise and mechanical motion[25, 26]; we take
the average weight of the two sidebands to cancel this ef-
fect and extract
〈
x
〉
2
(Supplementary Information). The
blue circles and dots in Fig.2b show the imprecision and
back-action versus
n
p
of each tone: as the imprecision
decreases, the fluctuations due to back-action increase,
increasing the mechanical occupation from 15 to 65 at
n
p
= 2
.
3
×
10
6
. In addition to the back-action, the
down-converted sideband has 5% more power than the
up-converted one (Fig.2a), which is consistent with the
expected asymmetry, and will be the subject of future
work. Both the back-action and the asymmetry observed
at
n
p
= 2
.
3
×
10
6
are consistent with small finite mi-
crowave occupation factor (
n
c
≈
0
.
6
±
0
.
1) in addition to
the quantum fluctuations.
In the BAE configuration (∆ = 0), the motional side-
bands overlap to become a single peak, and the re-
3
sponse is dramatically different. As the imprecision de-
creases, we do not observe a large increase in the me-
chanical fluctuations, as shown red dots in Fig.2b with
n
p
from both tones. The expected back-action into the
measured quadrature due to the finite sideband resolu-
tion is[6]:
〈
X
2
1
〉
ba
=
1
32
(
κ
ω
m
)
2
〈
X
2
2
〉
ba
≈
〈
X
2
2
〉
ba
/
700
≈
0
.
12
x
2
zp
at
n
p
= 4
.
7
×
10
6
. The measured back-action
of
〈
X
2
1
〉
ba
≈
10
x
2
zp
is likely due to ohmic heating of our
device. Nonetheless, we demonstrate avoidance of the
back-action noise by 10 dB at
n
p
= 2
.
3
×
10
6
. Most
importantly, we show that the back-action
〈
X
2
1
〉
ba
is 9
dB below the level set by quantum fluctuations of the
microwave field, 2(Γ
opt
/
Γ
m
)
x
2
zp
at
n
p
= 4
.
7
×
10
6
. The
quadrature imprecision is also below
x
zp
at this point:
〈
X
2
1
〉
imp
≈
0
.
6
x
2
zp
(Fig.2b, inset).
Even though our detection amplifier is far from
quantum-limited with a noise temperature of about 4
K, the strong BAE measurement results in a detector
noise product
√
S
X
1
S
F
of approximately 2.5 ̄
h
, lower
than other detection approaches with micro- and nano-
mechanical devices[27]. By minimizing device heating
due to circuit loss, we would expect this figure to drop
to
√
S
X
1
S
F
≈
0
.
3 ̄
h
, exceeding what is possible for a per-
fect measurement of position[27]:
√
S
x
S
F
= ̄
h/
2. With
a nearly quantum-limited amplifier[28], we expect even
further improvement reaching
√
S
X
1
S
F
≈
0
.
08 ̄
h
.
To show that the avoided back-action is indeed added
to the unobserved quadrature, we place a second set of
probe BAE tones, 20dB weaker than the pump BAE
tones at
n
p
= 1
.
1
×
10
6
(Fig.3a). We measure and
control the relative phase(
φ
) of these tones, and mea-
sure the resulting motional sidebands (Supplementary
Information). The signal from the probes measures a
quadrature variance along a rotated axis:
〈
X
(
φ
)
2
〉
=
〈
X
2
1
〉
cos
2
φ
+
〈
X
2
2
〉
sin
2
φ
. Figures 3b-c compare the sig-
nals from the two sets of BAE tones. As is apparent, the
fluctuations at
φ
=
π/
2, aligned along the
X
2
quadra-
ture, show maximal back-action heating.
We study the back-action noise into the
X
2
quadrature
(
〈
X
2
2
〉
ba
) versus microwave occupation factor by apply-
ing microwave noise to increase
n
c
. The HEMT ampli-
fier noise floor is carefully measured using a cryogenic
microwave switch, and this is the noise floor used in
measuring the increase in noise power ∆
η
in the elec-
trical resonator due to
n
c
(Supplementary Information).
Figure 3d shows the observed
〈
X
2
2
〉
ba
versus the in-
crease in the measured microwave noise power, where
〈
X
2
2
〉
ba
/x
2
zp
= 2(Γ
opt
/
Γ
m
)(2
n
c
+ 1) is predicted[6]. The
figure shows good fit to this formula, and since measured
microwave power is proportional to
n
c
, the slope of the
fit provides a calibration for
n
c
,
n
c
/
∆
η
= 0
.
22
±
0
.
02
(aW/Hz)
−
1
.
Most importantly, the fit intercept shown in Fig.3d of
1
.
1
±
0
.
1 shows excellent agreement to +1 expected from
quantum back-action, with the contribution of thermal
FIG. 2. Back-action evading measurement.
a
, An example
of measured motional sidebands in two-tone non-BAE (blue)
and BAE (red) configurations. The spectra clearly show re-
duced back-action in the BAE set-up. Inset, the blue area
under the Lorentzian peak of the motional sideband, which
is either the average of two sidebands in two-tone non-BAE
or the single sideband in BAE, represents the fluctuations
due to thermal (
〈
x
2
〉
th
)and measurement back-action forces
〈
x
2
〉
ba
(Supplementray Information). Imprecision (
〈
x
2
〉
imp
),
which is the additive noise inferred from the measurement
noise floor(
S
0
x
), is defined as the gray area under a Lorentzian
with its peak at
S
0
x
, and with linewidth Γ
m
.
b
, Measure-
ment imprecision(circles) and back-action(dots) of two-tone
non-BAE(blue) and BAE(red). The solid blue line represents
a fit to the measured back-action including classical noise in
the electrical resonator. The solid green line is the expected
quantum back-action from microwave shot noise. The mea-
sured back-action in BAE lies below the quantum back-action
above
n
p
≈
10
6
. The dashed blue line shows a fit to the mea-
sured imprecision and the dashed green line is the imprecision
expected at the quantum limit (= 1
/
(8Γ
opt
/
Γ
m
)). Inset, the
BAE imprecision reaches 0
.
6
x
2
zp
at
n
p
= 4
.
7
×
10
6
.
n
c
≈
0
.
1 (Supplementary Information). In this way, we
show that the mechanical device detects the quantum
fluctuations of the microwave field[29, 30].
These results lead the way towards manipulating the
quantum noise of a mechanical resonator. As described
in Ref. 6, in a particular run of the experiment, the mo-
tion is expected to be highly squeezed: we estimate the
4
FIG. 3. Measurement of back-action.
a
, Microwave set-up.
Strong BAE pumps(red) are applied symmetrically about the
microwave resonance (dashed line). Weak BAE probes(blue)
with a small detuning(+30 kHz) from the pumps are used
to measure the back-action from the BAE pumps. A weak
cooling tone with Γ
opt
≈
90 Hz is applied at the same time
(magenta).
b
, An example of measured mechanical fluctua-
tions along the BAE pump axis (red circles) and the probe
axis (blue circles).
φ
is the angle between these two axes.
The blue line is a fit to
A
sin
2
φ
+
B
. (Supplementary Infor-
mation).
c
, Polar plot of
b
. It defines
X
1
and
X
2
quadra-
tures along the direction of minimum and maximum fluctua-
tion, respectively.
d
, Mechanical fluctuations along the probe
axis at different microwave noise powers: ∆
η
= 5
.
7
,
9
.
2
,
13
,
21
aW/Hz (brown, green, blue, and red dots, respectively).
e
,
Back-action in the
X
2
quadrature normalized by quantum
back-action
〈
X
2
2
〉
qba
= 2(Γ
opt
/
Γ
m
)
x
2
zp
.
conditional squeezing
〈
X
2
1
〉
/
〈
X
2
2
〉
≈
0
.
01. However, af-
ter averaging over many runs from a thermal state, we
recover the thermal distribution on both quadratures and
lose squeezing. To produce a squeezed state from a ther-
mal state, feedback on the motion may be applied. Us-
ing a nearly quantum limited amplifier[28], we expect to
produce a squeezed state (
〈
X
2
1
〉
/x
2
zp
<
1) with
n
p
≈
10
5
,
which can be useful for detection of weak forces and fun-
damental studies of quantum decoherence[5]. We also
note that the mechanical mode reaching 7.2 mK demon-
strates a new application of a micro-mechanical resonator
as a primary ultra-low-temperature thermometer.
We would like to acknowledge Jared Hertzberg, Tris-
tan Rocheleau, Tchefor Ndukum, and Matt Shaw for
work on earlier experiments which lead to these results.
This work is supported by funding provided by the In-
stitute for Quantum Information and Matter, an NSF
Physics Frontiers Center with support of the Gordon and
Betty Moore Foundation (nsf-iqim 1125565), by DARPA
(DARPA-QUANTUM HR0011-10-1-0066), and by NSF
(nsf-dmr 1052647, nsf-eec 0832819).
∗
Correspondence and requests for materials should be ad-
dressed to Keith Schwab (email: schwab@caltech.edu).
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Supplementary Information for “Mechanically Detecting and
Avoiding the Quantum Fluctuations of a Microwave Field”
J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke,
P. Meystre, A. A. Clerk & K. C. Schwab
1 Sample fabrication
We start with a 525
μ
m thick
<
100
>
-oriented high resistivity (
>
10kΩ
·
cm) silicon wafer. After initial surface
preparation, a 100nm 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 photolithography followed by two-step wet etching
in Transene Al Etchant A and MF319. 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 150nm
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.
2 Theory
We start with a standard linearly coupled optomechanical Hamiltonian,
ˆ
H
=
~
(
ω
c
−
g
0
ˆ
x/x
zp
)
[
ˆ
a
†
ˆ
a
−〈
ˆ
a
†
ˆ
a
〉
(
t
)
]
+
~
ω
m
ˆ
c
†
ˆ
c
+
i
~
√
κ
L
( ̄
a
∗
in
(
t
)ˆ
a
−
̄
a
in
(
t
)ˆ
a
†
)
,
where ˆ
x
=
x
zp
(ˆ
c
+ ˆ
c
†
) is the mechanical position, ˆ
c
is the phonon annihilation operator, ˆ
a
is the photon
annihilation operator,
〈
ˆ
a
†
ˆ
a
〉
(
t
) is the classical mean photon number at time
t
, ̄
a
in
(
t
) is the driving field, and
all other constants are as defined in the text.
We make unitary transformations to a rotating, displaced frame, so that ˆ
a
=
e
−
iω
c
t
(
α
(
t
) +
ˆ
d
), with
α
(
t
) =
〈
ˆ
ae
iω
c
t
〉
, and so that
ˆ
b
= ˆ
ce
iω
m
t
. Including environmental dissipation, we arrive at the linearized
Heisenberg equations of motion,
̇
ˆ
d
=
−
ig
0
α
(
t
)
(
ˆ
be
−
iω
m
t
+
ˆ
b
†
e
iω
m
t
)
−
κ
2
ˆ
d
−
∑
j
=
L,R,int
√
κ
j
e
iω
c
t
ˆ
ξ
j
(
t
)
(2.1)
̇
ˆ
b
=
−
ig
0
ˆ
b
(
α
(
t
)
ˆ
d
†
+
α
∗
(
t
)
ˆ
d
)
−
Γ
m
0
2
ˆ
b
−
√
Γ
m
0
e
iω
m
t
ˆ
η
(
t
)
,
(2.2)
Here, the index
j
labels the three ports coupled to the cavity: the right and left coupling ports (
j
=
R,L
)
and a port representing internal cavity losses (
j
=
int
). The
ˆ
ξ
j
and ˆ
η
noise operators satisfy:
[
ˆ
ξ
j
(
t
)
,
ˆ
ξ
†
k
(
t
′
)] =
δ
j,k
δ
(
t
−
t
′
)
,
(2.3)
[ˆ
η
(
t
)
,
ˆ
η
†
(
t
′
)] =
δ
(
t
−
t
′
)
,
(2.4)
[
ˆ
ξ
j
(
t
)
,
ˆ
η
†
(
t
′
)] = 0
,
(2.5)
〈
ˆ
ξ
†
j
(
t
)
ˆ
ξ
j
(
t
′
)
〉
=
n
c,j
δ
(
t
−
t
′
)
,
(2.6)
〈
ˆ
η
†
(
t
)ˆ
η
(
t
′
)
〉
=
n
T
m
0
δ
(
t
−
t
′
)
,
(2.7)
1
arXiv:1312.4084v1 [quant-ph] 14 Dec 2013
where
n
c,j
and
n
T
m
0
are the thermal occupation factors associated with the cavity baths and mechanical
bath, respectively. Finally, the total cavity occupancy is defined as:
n
c
=
∑
j
=
L,R,int
κ
j
κ
n
c,j
(2.8)
2.1 Balanced detuned two-tone measurement
The first experimental measurement is performed using two balanced tones detuned by
±
(
ω
m
+ ∆) from
the microwave resonance, where we set ∆ = 5kHz
κ
. In addition, a third cooling tone is applied at
ω
c
−
ω
m
−
δω
cool
, whose sole purpose is to increase the mechanical linewidth from Γ
m
0
to Γ
m
= 100 Hz and
reduce the mechanical thermal occupancy. We set
δω
cool
= 35kHz
Γ
m
, which implies that the cooling
tone acts independently from the measurement tones.
The average microwave field amplitude takes the form (in our interaction picture)
α
(
t
) = ̄
a
cool
e
i
(
ω
m
+
δω
cool
)
+ ̄
a
pump
cos((
ω
m
+ ∆)
t
)
.
(2.9)
We define
n
p
=
|
̄
a
pump
|
2
/
4 and Γ
opt
=
4
g
2
0
κ
n
p
.
We solve the Heisenberg equations of motion, Eq. 2.1 and Eq. 2.2, and calculate symmetrized noise
spectral densities defined as,
S
q
(
ω
)
≡
1
2
∫
∞
−∞
〈
ˆ
q
(
t
)ˆ
q
†
(0) + ˆ
q
†
(0)ˆ
q
(
t
)
〉
e
iωt
dt,
(2.10)
for an operator ˆ
q
.
Ignoring small bad-cavity corrections (which are small as (
κ/ω
m
)
2
), the noise spectral density of mechan-
ical motion is found to be
S
x
(
ω
) =
x
2
zp
Γ
m
ω
2
+ Γ
2
m
/
4
(1 + 2
n
T
m
+ 2
n
ba
)
,
(2.11)
where Γ
m
is the enhanced mechanical linewidth (i.e. it includes the optical damping associated with the
cooling tone),
n
T
m
is the cavity-cooled mechanical thermal occupancy, and
n
ba
= (Γ
opt
/
Γ
m
)(2
n
c
+ 1) is the
additional heating due to the measurement tones. It thus follows that
〈
x
2
〉
=
x
2
zpt
(1 + 2 ̄
n
m
), where the
effective mechanical thermal occupancy is ̄
n
m
=
n
T
m
+
n
ba
.
The output spectrum of the up-converted sideband (centered at
ω
c
−
∆ in the lab frame) is,
S
red
R
(
ω
+ ∆) =
1
2
+
n
c,R
+
4
κ
R
κ
(
n
c
−
n
c,R
) +
κ
R
κ
Γ
opt
Γ
m
ω
2
+ Γ
2
m
/
4
( ̄
n
m
+
n
c,R
−
2
n
c
)
,
(2.12)
where we have assumed
|
ω
+ ∆
|
κ
. Similarly, the output spectrum of the down-converted sideband
(centered at
ω
c
+ ∆ in the lab frame) is,
S
blue
R
(
ω
−
∆) =
1
2
+
n
c,R
+
4
κ
R
κ
(
n
c
−
n
c,R
) +
κ
R
κ
Γ
opt
Γ
m
ω
2
+ Γ
2
m
/
4
( ̄
n
m
+ 1
−
n
c,R
+ 2
n
c
)
,
(2.13)
where we take
|
ω
−
∆
|
κ
We see that the total weight of the two mechanically-induced sidebands in the output spectrum have
information on both the mechanical occupancy ̄
n
m
as well as the cavity thermal occupancy
n
c
and the
amount of thermal cavity noise incident from the right port,
n
c,R
. Note that the cavity thermal noises affect
the sidebands in an asymmetric manner. By taking the average of the two sideband spectra and then fitting
to a Lorentzian, we can extract ̄
n
m
without needing to independently measure
n
c
,
n
c,R
.
2.2 Back-action evading measurement
Our experiment differs from previous works and analysis [1] as four drive tones are used: two principle BAE
tones that allow for a BAE measurement of the
ˆ
X
1
mechanical quadrature by monitoring the cavity output at
resonance, and two weaker tones which allows one to simultaneously (and weakly) probe a second mechanical
quadrature
ˆ
X
φ
using microwaves slightly detuned a frequency
δ
κ
from the cavity resonance. Similar to
the detuned two-tone measurements described in Sec. 2.1, we also employ a fifth cooling tone detuned from
the red mechanical sideband; similar to that case, this tone only serves to increase the mechanical linewidth
and reduce its thermal occupancy. Our analysis below also includes the impact of cavity thermal noise.
The average microwave fields due to the four measurement tones and one cooling tone thus takes the
form:
α
(
t
) = ̄
a
pump
cos
ω
m
t
+ ̄
a
probe
e
−
iδt
cos(
ω
m
t
+
φ
) + ̄
a
cool
e
i
(
ω
m
+
δω
cool
)
(2.14)
where
δ
represents the offset in the centre frequency of the probe tones versus the principle BAE measurement
tones (the “pump” tones). The presence of both sets of measurement tones leads to a small deviation
from a perfect QND interaction Hamiltonian. We define the interaction strengths
G
pump
=
g
0
̄
a
pump
/
2,
G
probe
=
g
0
̄
a
probe
/
2. The linearized optomechanical interaction Hamiltonian (in an interaction picture with
respect to the bare cavity and mechanical Hamiltonian) takes the form
ˆ
H
int
=
1
x
zp
[
G
pump
(
ˆ
d
+
ˆ
d
†
)
ˆ
X
1
+
G
probe
(
ˆ
de
−
iδt
+
ˆ
d
†
e
iδt
)
(
cos
φ
ˆ
X
1
−
sin
φ
ˆ
X
2
)]
(2.15)
where we have neglected writing the interaction with the cooling tone. The quadrature operators are defined
as:
ˆ
X
1
=
x
zp
(
ˆ
b
+
ˆ
b
†
)
,
ˆ
X
2
=
−
ix
zp
(
ˆ
b
−
ˆ
b
†
)
(2.16)
ˆ
X
φ
=
x
zp
(
ˆ
be
iφ
+
ˆ
b
†
e
−
iφ
)
= cos
φ
ˆ
X
1
−
sin
φ
ˆ
X
2
(2.17)
The
G
probe
term in Eq. (2.15) describes the measurement of the
ˆ
X
φ
quadrature by the cavity field at
ω
=
δ
(in the interaction picture); it also induces a backaction on the
ˆ
X
1
quadrature measured by the main
measurement tones (the “pump” tones). By taking ̄
a
probe
sufficiently small and
|
δ
|
Γ
m
, this additional
backaction can be minimal, as we show below.
We first establish that the cavity output spectrum near
ω
= 0 (near
ω
=
δ
) has information on the
mechanical
X
1
(
X
φ
) quadrature. This follows immediately from the Fourier-transformed equation of motion
for the cavity field:
ˆ
d
[
ω
] =
1
−
iω
+
κ/
2
−
i
G
pump
x
zp
ˆ
X
1
[
ω
]
−
i
G
probe
x
zp
ˆ
X
φ
[
ω
−
δ
]
−
∑
j
=
L,R,int
√
κ
j
ˆ
d
j,in
(2.18)
As the mechanical quadrature operators
ˆ
X
j
[
ω
] are only appreciably non-zero for
|
ω
|
.
Γ
m
, and as
δ
Γ
m
,
we see that
ˆ
d
[
ω
'
0] only contains information about
X
1
, while
ˆ
d
[
ω
'
δ
] only contains information on
X
φ
.
Using the standard input-output relation
ˆ
d
j,out
(
t
) =
ˆ
d
j,in
(
t
)+
√
κ
j
ˆ
d
(
t
), we see that this statement also carries
over to the measured output field leaving the right port,
ˆ
d
R,out
[
ω
].
We now turn to the additional backaction effect associated with the probe tones. We solve Eqs. (2.1)
and (2.2) to leading order in the probe-tone amplitudes. As the Hamiltonian deviates from the perfect QND
limit, there is now a weak additional optical damping of the mechanical quadratures. In the limit of interest
(
|
δ
|
Γ
m
), these additional damping terms Γ
extra
scale as:
Γ
extra
∝
Γ
m
Γ
2
opt
δ
2
(
̄
a
probe
̄
a
pump
)
2
(2.19)
For the parameters of our measurement, this additional damping rate is approximately 10
−
4
times smaller
than Γ
m
, and thus plays no role.
Ignoring this non-QND damping effect, we can easily calculate the noise spectra of the mechanical quadra-
tures, again considering the effects of the probe tones to leading order. Consider first the
ˆ
X
2
quadrature,
which is heating by the backaction of the main (pump-tone) quadrature measurement. We find
S
X
2
(
ω
)
/x
2
zp
=
Γ
m
ω
2
+ (Γ
m
/
2)
2
(
1 + 2
n
T
m
+ 2
n
BAE
ba
)
,
(2.20)