of 17
Observation and Interpretation of Motional Sideband Asymmetry in a Quantum
Electromechanical Device
A. J. Weinstein,
1,2
C. U. Lei,
1,2
E. E. Wollman,
1,2
J. Suh,
1,2
,
A. Metelmann,
3
A. A. Clerk,
3
and K. C. Schwab
1,2
,*
1
Applied Physics, Caltech, Pasadena, California 91125, USA
2
Kavli Nanoscience Institute, Caltech, Pasadena, California 91125, USA
3
Department of Physics, McGill University, Montreal, Quebec H3A 2T8, Canada
(Received 15 May 2014; revised manuscript received 31 July 2014; published 7 October 2014)
Quantum electromechanical systems offer a unique opportunity to probe quantum noise properties in
macroscopic devices, properties that ultimately stem from Heisenberg
s uncertainty relations. A simple
example of this behavior is expected to occur in a microwave parametric transducer, where mechanical
motion generates motional sidebands corresponding to the up-and-down frequency conversion of
microwave photons. Because of quantum vacuum noise, the rates of these processes are expected to
be unequal. We measure this fundamental imbalance in a microwave transducer coupled to a radio-
frequency mechanical mode, cooled near the ground state of motion. We also discuss the subtle origin of
this imbalance: depending on the measurement scheme, the imbalance is most naturally attributed to the
quantum fluctuations of either the mechanical mode or of the electromagnetic field.
DOI:
10.1103/PhysRevX.4.041003
Subject Areas: Mechanics, Quantum Physics,
Quantum Information
I. INTRODUCTION
A fascinating aspect of quantum measurement is that the
outcome of experiments and the apparent nature of the object
under study depend critically on the properties of both the
system and the measurement scheme
[1]
. An excellent
illustration is found when considering measurements of
the quantum harmonic oscillator. When measured with an
ideal energy detector, the observed signals will demonstrate
energy-level quantization
[2,3]
; however, if measured
instead with an ideal position detector, no evidence of
quantized energy levels is found and the measured signals
appear to be those of a very cold, classical oscillator
[4,5]
.
The details of the measurement are as essential to the
apparent nature of the system under study as the properties
of the system itself
succinctly expressed by Roy Glauber,
A photon is what a photodetector detects
[6]
.
To describe the measured noise of quantum systems, it is
often useful to make use of so-called quantum noise
spectral densities, which, in general, are not symmetric
functions of frequency:
S
xx
ð
ω
Þ
S
xx
ðþ
ω
Þ
, where
S
xx
ð
ω
Þ
is the spectral density of the observable
x
ð
t
Þ
, defined as the
Fourier transform of
h
ˆ
x
ð
t
Þ
ˆ
x
ð
0
Þi
[7]
. For a quantum har-
monic oscillator, the negative- and positive-frequency sides
of this spectral density describe the ability of the system to
emit or absorb energy. In the ground state, there is no ability
for the harmonic oscillator to emit energy, so that
S
xx
ð
ω
m
Þ¼
0
. It can, however, absorb energy, and as a
result,
S
xx
ðþ
ω
m
Þ¼ð
4
=
γ
m
Þ
x
2
ZP
, where
x
ZP
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
=
2
m
ω
m
p
is the amplitude of zero-point fluctuations for a mechanical
oscillator with mass
m
, resonance frequency
ω
m
, and
damping rate
γ
m
. More generally, for a mechanical oscil-
lator in a thermal state with an occupation factor
̄
n
m
, the
spectral densities follow
S
xx
ð
ω
m
Þ¼ð
4
=
γ
m
Þ
x
2
ZP
̄
n
m
and
S
xx
ðþ
ω
m
Þ¼ð
4
=
γ
m
Þ
x
2
ZP
ð
̄
n
m
þ
1
Þ
[3]
. This asymmetric-
in-frequency motional noise spectrum was first measured
in atomic systems prepared in quantum ground states of
motion
[8
10]
, where the motional sideband absorption
and fluorescence spectra were detected via photodetection.
Analogous quantum noise effects can also be studied in
macroscopic mechanical systems, using electromechanical
and optomechanical devices prepared and probed at quan-
tum limits
[5,11
13]
. These systems exhibit the Raman-
like processes of the up- and down-conversion of photons,
resulting from the parametric coupling between mechanical
motion and electromagnetic modes of a resonant cavity; the
rates of these processes should naturally mirror the asym-
metry in the mechanical quantum noise spectral density
S
xx
ð
ω
m
Þ
. Recent experiments in optomechanics have
demonstrated this expected imbalance between up- and
down-converted sidebands
[14,15]
. Here, we demonstrate
the analogous physics in a quantum circuit, where it is
now microwave photons (not optical photons) that probe
the mechanical motion.
We also address a subtlety about these measurements
that originates from their use of linear detection of the
scattered electromagnetic field: they measure the field
*
To whom all correspondence should be addressed.
schwab@caltech.edu
Present address: Korea Research Institute of Standards and
Science, Daejeon 305-340, Republic of Korea.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 3.0 License
. Further distri-
bution of this work must maintain attribution to the author(s) and
the published article
s title, journal citation, and DOI.
PHYSICAL REVIEW X
4,
041003 (2014)
2160-3308
=
14
=
4(4)
=
041003(17)
041003-1
Published by the American Physical Society
amplitude (e.g., via heterodyne detection). This scheme is
in contrast to measurements employing direct photodetec-
tion, where one filters the output light and counts photons
associated with a motional sideband. Although the pre-
dicted and measured motional sideband asymmetries
obtained using either detection method are identical
[14
16]
, the interpretation is more nuanced when one
employs linear field detection. As discussed by Khalili
et al.
[16]
, the asymmetry in this case can be fully attributed
to the detector, namely, the presence of a precisely tuned
correlation between the backaction noise generated by the
measurement device and its imprecision noise (see
Appendix
B
). We provide a simple exposition of this
physics using standard input-output theory, which lets us
easily track the scattering of incident vacuum fluctuations.
In the case of linear detection of the cavity output field, the
imbalance is naturally attributed to the input electromag-
netic field fluctuations (classical and quantum); the intrinsic
quantum fluctuations of the mechanical mode contribute
equally to the up- and down-converted spectra. In contrast,
in experiments that employ direct photodetection, the
asymmetric quantum noise of the mechanical motion
directly contributes to the imbalance of the output spectrum
(in addition to any contribution from excess classical
fluctuations in the input electromagnetic fields). After a
brief discussion of these theoretical issues, we present
measurements of the imbalance in a microwave-frequency
electromechanical device.
II. THEORY
We begin with the Hamiltonian of our electromechanical
system
ˆ
H
¼
ω
c
ˆ
a
ˆ
a
þ
ω
m
ˆ
b
ˆ
b
þ
g
0
ˆ
a
ˆ
a
ð
ˆ
b
þ
ˆ
b
Þ
;
ð
1
Þ
where
ˆ
a
ð
ˆ
a
Þ
is the annihilation (creation) operator of the
microwave resonator mode with frequency
ω
c
,
ˆ
b
ð
ˆ
b
Þ
is the
annihilation (creation) operator of the mechanical resonator
with frequency
ω
m
, and
g
0
is the parametric coupling
strength between the two modes.
We consider the standard regime of a cavity strongly
driven at frequency
ω
p
, where dissipation is treated as per
standard input-output theory
[17]
; we also consider a two-
sided cavity, which corresponds to our experimental setup.
Writing the cavity and mechanical fields in terms of
their classical and quantum parts
ˆ
a
¼
e
i
ω
p
t
ð
̄
a
þ
ˆ
d
Þ
and
ˆ
b
¼
̄
b
þ
ˆ
c
, we linearize to obtain the following
Heisenberg-Langevin equations:
_
ˆ
d
¼

i
Δ
þ
κ
2

ˆ
d
iG
ð
ˆ
c
þ
ˆ
c
Þ
X
σ
¼
R;L
ffiffiffiffiffi
κ
σ
p
ˆ
d
σ
;
in
;
_
ˆ
c
¼

i
ω
m
þ
γ
m
2

ˆ
c
iG
ð
ˆ
d
þ
ˆ
d
Þ
ffiffiffiffiffi
γ
m
p
ˆ
c
in
;
where
Δ
¼
ω
c
ω
p
,
G
¼
g
0
j
̄
a
j
, and
κ
¼
κ
L
þ
κ
R
(
γ
m
)is
the microwave- (mechanical-) resonator damping rate. The
operators
ˆ
d
σ
;
in
ð
t
Þ
and
ˆ
c
in
ð
t
Þ
describe noise incident on
the microwave and mechanical resonator, respectively, and
satisfy
h
ˆ
d
σ
;
in
ð
t
Þ
ˆ
d
σ
0
;
in
ð
t
0
Þi¼ð
n
th
σ
þ
α
Þ
δ
σ
;
σ
0
δ
ð
t
t
0
Þ
;
h
ˆ
d
σ
;
in
ð
t
Þ
ˆ
d
σ
0
;
in
ð
t
0
Þi ¼
n
th
σ
δ
σ
;
σ
0
δ
ð
t
t
0
Þ
;
h
ˆ
c
in
ð
t
Þ
ˆ
c
in
ð
t
0
Þi¼ð
n
th
m
þ
β
Þ
δ
ð
t
t
0
Þ
;
h
ˆ
c
in
ð
t
Þ
ˆ
c
in
ð
t
0
Þi ¼
n
th
m
δ
ð
t
t
0
Þ
:
Here,
n
th
m
(
n
th
σ
) denotes the amount of thermal fluctuations
incident on the mechanical resonator (microwave resonator
from port
σ
), and
α
and
β
describe the quantum vacuum
fluctuations driving the microwave and mechanical reso-
nators, respectively; we have
α
¼
β
¼
1
, consistent with
the uncertainty relations and the canonical commutation
relations of the noise operators. In what follows, we keep
α
and
β
unspecified in order to clearly track the contributions
of both mechanical and electromagnetic vacuum noise to
the measured noise spectrum.
We include multiple bath temperatures (
n
th
σ
) to describe
the various sources of heating in microwave circuits.
Compared to optical cavities which are passively cooled
well into the ground state (
<
10
4
K), microwave cavities can
have significant thermal occupation even at temperatures
reached in the dilution refrigerator. Filtering on the input
and output transmission lines suppresses incident room-
temperature noise; however, other issues may remain, like
internal dissipation in the cavity
[18]
or thermal noise from
refrigerator components. Additionally, there are other
issues common to both microwave and optical systems,
such as source-phase noise
[19]
and cavity-frequency jitter
[20]
. Whatever the source, noise in the system can be
generalized into two categories based on how the noise
contributes to the measured signal, either by radiating
directly into the cavity (
n
th
L
) or by radiating into both the
cavity and detector (
n
th
R
). For this experiment,
n
th
R
describes
noise generated from a hot circulator on the output line
while
n
th
L
describes all other significant noise sources.
We further specialize to the case where a single micro-
wave-cavity drive is applied at
ω
p
¼
ω
c
Δ
with
Δ
either

ω
m
, and consider the up- and down-converted sidebands
generated by the mechanical motion. For simplicity,
we ignore any internal loss of the cavity, consider the
system to be in the sideband-resolved regime (
κ
ω
m
),
and also consider the limit of a weak cooperativity,
γ
opt
¼
4
G
2
=
κ
γ
m
. This last condition implies that the
backaction effects on the mechanics are minimal: the
mechanical linewidth and temperature are set by its
coupling to its intrinsic dissipative bath.
There are several ways one could now measure the
outgoing field
ˆ
d
R;
out
from the cavity and the corresponding
A. J. WEINSTEIN
et al.
PHYS. REV. X
4,
041003 (2014)
041003-2
power spectrum of its fluctuations. One general approach is
to first measure the time-dependent quadrature amplitudes
of the output field and then use these elements to calculate a
power spectrum. We do this in our experiment by using a
linear amplifier to measure the voltage associated with the
outgoing field. In optics setups, it can be instead done using
heterodyne detection, where one interferes the outgoing
field with a reference beam. (This approach was used in
Refs.
[14,15]
.) In both cases, one calculates the power
spectrum from classical representations of quadrature
amplitudes and is thus equally sensitive to absorption
and emission of photons from this field. One thus neces-
sarily measures a symmetrized amplitude power spectral
density, which here takes the form
̄
S
II;
tot
½
ω
1
2
Z
dt
h
ˆ
I
tot
ð
t
Þ
ˆ
I
tot
ð
0
Þþ
ˆ
I
tot
ð
0
Þ
ˆ
I
tot
ð
t
Þi
e
i
ω
t
;
ð
2
Þ
with the amplitude of the output field
ˆ
I
tot
¼
ˆ
d
R;
out
þ
ˆ
d
R;
out
and where
ˆ
d
R;
out
¼
ˆ
d
R;
in
þ
ffiffiffiffiffi
κ
R
p
ˆ
d
. The output spectrum
near the cavity resonance for the two choices of drive
detuning are found to be
̄
S
II;
tot
½
ω
j
Δ
¼þ
ω
m
¼
̄
S
0
þ
κ
R
κ
γ
opt
γ
m
ð
γ
m
2
Þ
2
þð
ω
ω
c
Þ
2
×

n
th
m
þ
β
2


n
th
eff
þ
α
2

;
ð
3
Þ
̄
S
II;
tot
½
ω
j
Δ
¼
ω
m
¼
̄
S
0
þ
κ
R
κ
γ
opt
γ
m
ð
γ
m
2
Þ
2
þð
ω
ω
c
Þ
2
×

n
th
m
þ
β
2

þ

n
th
eff
þ
α
2

;
ð
4
Þ
where for
Δ
¼
ω
m
(
Δ
¼
ω
m
), the up- (down-) converted
sideband is centered on the cavity resonance. The
noise floor for both cases is given by
̄
S
0
¼
α
=
2
þ
n
th
R
þ
4
κ
R
ð
n
th
c
n
th
R
Þ
=
κ
, and we have defined
n
th
eff
¼
2
n
th
c
n
th
R
[where
n
th
c
¼ð
κ
L
n
th
L
þ
κ
R
n
th
R
Þ
=
κ
is the effective cavity
thermal occupancy]. In Fig.
1(c)
, we illustrate the under-
lying components of this spectrum.
One sees explicitly that the sideband imbalance
̄
S
II;
tot
½
ω
j
Δ
¼
ω
m
̄
S
II;
tot
½
ω
j
Δ
¼þ
ω
m
is proportional to
(
2
n
th
eff
þ
α
) and hence is entirely due to fluctuations in
the microwave fields driving the cavity. This interpretation
is true both when this noise is thermal and when it is purely
quantum (i.e.,
n
th
R
¼
n
th
L
¼
0
). These terms in the spectrum
result from the interference between the two ways the
incident field noise can reach the output: either by directly
being transmitted through the cavity or by first driving the
mechanical resonator whose position then modulates
the amplitude quadrature of the outgoing microwaves.
(See Appendix
A
for further insights based on a scattering
approach.) This interference is the basic mechanism of
noise squashing, which in the case of thermal noise was
previously observed in a microwave-cavity-based electro-
mechanical system
[4]
. This mechanism can also be fully
described using a general linear measurement forma-
lism
[16]
, where it is attributed to the presence of
correlations between the backaction and imprecision
noises of the detector, correlations that are out of phase
and have magnitude
=
2
in the zero-temperature limit.
Interestingly, this precise value plays a special role in
(a)
(b)
(c)
FIG. 1. Comparison between linear detection and photodetec-
tion. (a) Pump scheme. We consider a single microwave cavity
(dotted line) pumped at
ω
c
ω
m
þ
δ
Þ
(green bars). The up-
converted (red bar) and down-converted (blue bar) motional
sidebands are placed tightly within the cavity linewidth. For
figure clarity, the occupations of the microwave and mechanical
modes are assumed to be zero. (b) Linear detection. The quantum
contribution from the symmetrized motional noise
̄
S
xx
is present
in both sidebands. Microwave shot noise (brown band) and
amplifier noise (beige band) combine to form the imprecision
noise
̄
S
II
. This measurement is sensitive to noise correlation
between the microwave and mechanical modes (
̄
S
IF
), which
results in asymmetric squashing (red region) and antisquashing
(blue region) of the noise floor. (c) Photodetection. Normal-
ordered detection is sensitive to the asymmetric motional noise
spectrum
S
xx
. The detector is not sensitive to microwave shot
noise, and the noise floor (
S
II
) is from detector nonidealities
(beige band), analogous to dark counts for a photodetector.
Although the source is different, the sideband imbalance is
identical in both photodetection and linear detection. For math-
ematical descriptions of
S
II
and
S
IF
, refer to Appendix
B
.
OBSERVATION AND INTERPRETATION OF MOTIONAL
...
PHYS. REV. X
4,
041003 (2014)
041003-3
the theory of quantum limits on linear amplification
[7]
.
(See Appendix
B
for more details.)
The above calculation also shows that both thermal and
zero-point force noise emanating from the mechanical bath
(i.e., terms
n
th
m
þ
β
=
2
) contribute symmetrically to
Eqs.
(3)
and
(4)
and hence play no role in determining
the asymmetry of the sidebands. In the weak-cooperativity
limit, it is the mechanical bath that almost entirely deter-
mines the mechanical oscillator fluctuations. This suggests
that the sideband asymmetry observed using linear detection
of the scattered field is not directly probing the asymmetric
quantum noise spectrum of the mechanical mode.
An alternate measurement strategy to amplitude detec-
tion [which leads to Eq.
(2)
] is to first filter the output signal
to a narrow bandwidth around a frequency
ω
and then
perform direct photodetection. One is thus measuring
power directly without first measuring field amplitudes,
and in a manner that is only sensitive to the absorption of
photons. As a result, such a measurement is described by
the normal-ordered spectrum
S
N
II;
tot
½
ω
Z
dt
h
ˆ
d
R;
out
ð
0
Þ
ˆ
d
R;
out
ð
t
Þi
e
i
ω
t
;
ð
5
Þ
with output spectra given by
S
N
II;
tot
½
ω
j
Δ
¼þ
ω
m
¼

̄
S
0
α
2

þ
κ
R
κ
γ
opt
γ
m
ð
γ
m
2
Þ
2
þð
ω
ω
c
Þ
2
ð
n
th
m
n
th
eff
Þ
;
ð
6
Þ
S
N
II;
tot
½
ω
j
Δ
¼
ω
m
¼

̄
S
0
α
2

þ
κ
R
κ
γ
opt
γ
m
ð
γ
m
2
Þ
2
þð
ω
ω
c
Þ
2
ð
n
th
m
þ
β
þ
n
th
eff
Þ
:
ð
7
Þ
Note that when one sets
α
¼
β
¼
1
, the asymmetry of these
normal-ordered spectra
S
N
II;
tot
½
ω
j
Δ
¼
ω
m
S
N
II;
tot
½
ω
j
Δ
¼þ
ω
m
is identical to that obtained from the linear measurement
[where spectra are calculated using Eq.
(2)
]. In this case,
however, the asymmetry is naturally attributed to both the
mechanical quantum fluctuations
β
and to the thermal
microwave fluctuations described by
n
th
eff
; this is illustrated
in Fig.
1(b)
. Note that in ideal direct photodetection (i.e., no
incident thermal fluctuations on the cavity), one cannot
attribute the zero-temperature sideband asymmetry to a
correlation between backaction-driven position fluctua-
tions and imprecision noise, as there is no imprecision
noise floor.
While the above simple calculations suggest that the
sideband asymmetries measured using linear detection
versus direct photodetection have different origins, it is
no accident that the magnitudes of the asymmetry are the
same in both schemes. This follows directly from the fact
that the canonical commutation relation of the output field
is the same as the input field
½
ˆ
d
R;
out
ð
ω
Þ
;
ˆ
d
R;
out
ð
ω
0
Þ ¼
αδ
ð
ω
þ
ω
0
Þ
. It necessarily follows that the spectra in
Eqs.
(2)
and
(5)
will differ only by a frequency-independent
noise floor of magnitude
α
=
2
[16,19]
. If one assumes this
commutation relation, then one can legitimately say that
both spectra essentially measure the same thing. However,
on a formal level, this reasoning involves an additional
assumption on the value of
β
:if
β
α
, then the output
commutator would not be the same as the input commu-
tator; see Appendix
A
.
Having explored the interpretation subtleties associated
with sideband asymmetry, we now turn to presenting our
main result: the experimental observation of this imbalance
in a microwave-cavity-based electromechanical system.
III. RESULTS
Our system is composed of a superconducting
microwave resonator, also referred to as a
cavity,
where
the resonance frequency is modulated by the motion of a
compliant membrane
[13]
. This frequency modulation
leads to the desired parametric coupling between
microwave field and mechanical motion [Fig.
2(a)
].
Measurements of the cavity response below 100 mK yield
the resonance frequency
ω
c
¼
2
π
×
5
.
4
GHz, total loss
rate
κ
¼
2
π
×
860
kHz, output coupling rate
κ
R
¼
2
π
×
450
kHz, and input coupling rate
κ
L
¼
2
π
×
150
kHz. The
capacitor top gate is a flexible aluminum membrane
(
40
μ
40
μ
150
nm) with a fundamental drumhead
mode with resonance frequency
ω
m
¼
2
π
×
4
.
0
MHz and
intrinsic loss rate
γ
m
¼
2
π
×
10
Hz at 20 mK. Motional
displacement of the top gate modulates the microwave-
resonance frequency with an estimated coupling rate
of
g
0
¼ð
ω
c
=
x
Þ
x
ZP
¼
2
π
×
16
Hz.
In Fig.
2(c)
, we present a schematic of the measurement
circuit. Tunable cavity filters at room temperature reduce
the source-phase noise to the thermal noise level at 300 K;
cryogenic attenuators further reduce the noise down to the
shot noise level
[4]
. A pair of microwave switches at the
device stage selects between the device or a bypass
connection for high precision noise floor calibration of
the cryogenic amplifier. The output signal passes through
two cryocirculators at approximately 100 mK, followed by
a cryogenic high-electron-mobility transistor amplifier
(HEMT) at 4.2 K, and finally to room-temperature circuits
for analysis. The occupation factor of the microwave
resonator
n
th
c
, which is expected to thermalize below
5
×
10
3
at temperatures below 50 mK, can be increased
and controlled by the injection of microwave-frequency
noise from amplified room-temperature Johnson noise.
From careful measurements of the noise power emanating
from the cavity at zero pumping compared to power spectra
with the bypass switched in place, we conclude that there is
a small contribution to
n
th
c
due to thermal radiation from the
A. J. WEINSTEIN
et al.
PHYS. REV. X
4,
041003 (2014)
041003-4
isolated port of the cryogenic circulators, given by the
occupation factor
n
th
R
¼
0
.
25

0
.
03
[21]
. Since performing
this experiment, we have reduced the temperature of the
circulator and have decreased the output-port occupation
below measurement sensitivity levels (
n
th
R
0
.
03
).
When a single microwave tone is applied to the device at
ω
p
, the parametric coupling converts mechanical oscilla-
tions at
ω
m
to up- and down-converted sidebands at
ω
p

ω
m
. In this experiment, we apply microwave tones
at frequencies near
ω
c

ω
m
and at powers given by the
mean number of photons in the resonator
n
p
. The micro-
wave resonance suppresses motional sidebands outside of
the linewidth, and we consider only the contributions of
signals converted to frequencies near
ω
c
. These sidebands
are the Lorentzian components of the noise spectra of
Eqs.
(3)
and
(4)
, which for the remainder of the paper are
denoted by plus and minus signs, respectively.
Throughout the measurement, we simultaneously apply
three microwave tones. We place a cooling tone at
ω
c
ω
m
δ
c
to control the effective mechanical damping rate
γ
M
and mode occupation
̄
n
m
via backaction cooling
[22,23]
.
Two additional probe tones, placed at
ω
c
ω
m
þ
δ
Þ
,
produce up- and down-converted sidebands symmetrically
detuned from the cavity center [Fig.
3(a)
]. The detunings are
chosen to ensure no interference between the sidebands
(
δ
c
¼
2
π
×
30
kHz,
δ
¼
2
π
×
5
kHz) so that we may con-
sider the probe sidebands as independent measurements of
the dressed mechanical mode.
There are several differences between the theory model
developed above and the experimental realization described
here. In practice, we measure the mechanical sidebands
produced in a two-port microwave resonator with limited
sideband resolution and a noisy output port, and in the
presence of multiple injected tones with a range of
detunings and powers. From further analysis of the exper-
imental configuration (see Appendix
A2
), we estimate
corrections to Eqs.
(3)
and
(4)
that are
1
and far below
the measurement resolution of our system.
To convert the motional sideband powers into equivalent
mechanical occupation
ð
̄
n
m
Þ
, we turn off the cooling tone
and measure the probe sidebands (
δ
¼
2
π
×
500
Hz) with
low optical damping (
n
þ
p
¼
n
p
5
×
10
2
) and high
mechanical occupation set by the cryostat temperature.
Such low pump powers ensure that the mechanical side-
band signals are dominated solely by the intrinsic thermal
noise of the mechanical mode; other effects such as
dynamical backaction, cavity-noise interference, or
mechanical bath heating are negligible in this regime
[20,21]
. Regulating the temperature to calibrated levels
between 20 and 200 mK, we calculate the integrated noise
(a)
(b)
(c)
Red probe
Blue probe
Filter cavity
Noise injection
analyzer
FIG. 2. Device, calibration, and measurement scheme. (a) Electron micrograph of the measured device. A suspended aluminum (gray
areas) membrane patterned on silicon (blue background) forms the electromechanical capacitor. It is connected to the surrounding spiral
inductor to form a microwave resonator. Out of view, coupling capacitors on either side of the inductor couple the device to input and
output coplanar waveguides. (b) Motional sideband calibration. The cryostat temperature is regulated while the mechanical mode is
weakly probed with microwave tones set at
ω
c
þ
ω
m
þ
δ
(blue) and at
ω
c
ω
m
δ
(red), with detunings
δ
¼
2
π
×
500
Hz. The
observed linear dependence provides the calibration between the normalized sideband power and the mechanical occupation factor.
Insets: Up-converted motional sideband spectra collected at 20 mK (top) and 200 mK (bottom), with
Δ
ω
¼
ω
ð
ω
c
δ
Þ
. (c) Schematic
of the microwave-measurement circuit.
OBSERVATION AND INTERPRETATION OF MOTIONAL
...
PHYS. REV. X
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041003 (2014)
041003-5
power under the sideband Lorentzians
P

m
, normalized by
the respective microwave-probe power transmitted through
the device
P

thru
. In the limit of high thermal occupation, the
normalized power is directly proportional to
̄
n
m
[24]
.Aswe
vary the cryostat temperature
T
, we compare the normalized
power to the thermal occupation factor
½
exp
ð
ω
m
=k
B
T
Þ
1

1
[Fig.
2(b)
]. A linear fit yields the conversion factors for the
up-converted (
n
þ
m
) and down-converted (
n
m
) sidebands:
n
þ
m
¼ð
9
.
9

0
.
2
Þ
×
10
8
P
þ
m
=P
þ
thru
and
n
m
¼ð
5
.
4

0
.
1
Þ
×
10
8
P
m
=P
thru
. The factor of approximately 2 between cali-
bration factors at the two pump detunings is due to the
presence of a parasitic bypass channel in the microwave
circuit that allows pump signals toweakly transmit across the
input and output ports of the device while completely
bypassing the microwave resonator. (See Sec. 4 of Ref.
[21]
.)
Further detuning the probe tones (
δ
¼
2
π
×
5
kHz) and
turning on the cooling tone (
δ
c
¼
2
π
×
30
kHz), we
explore the sideband ratio
n
m
=n
þ
m
over various mechanical
and microwave occupations. To reduce
̄
n
m
to values
approaching 1, we increase the cooling-tone power up to
n
cool
p
¼
4
×
10
5
. For sideband characterization, the probe-
tone powers are set to
n
p
¼
n
þ
p
¼
10
5
in order to achieve
sufficient measurement sensitivity and the probe sideband
spectra are analyzed using the conversion factors described
above. The imbalance between
n
m
and
n
þ
m
is clearly evident
in the noise spectra [Fig.
3(b)
].
As further demonstration of the asymmetry with respect
to
n
th
eff
, we plot the ratio
n
m
=n
þ
m
as a function of
n
þ
m
in
Fig.
3(c)
. Each curve corresponds to one setting of injected
microwave noise. The data show excellent agreement to the
expected ratio
n
m
=n
þ
m
¼
1
þð
2
n
th
eff
þ
1
Þ
=n
þ
m
. This relation-
ship highlights the combined effect of quantum and
classical noise in Eqs.
(3)
and
(4)
. By fitting each curve
to a two-parameter model
a
þ
b=n
þ
m
, we find an average
constant offset
a
¼
0
.
99

0
.
02
for all curves, accurately
matching the model and confirming our calibration tech-
niques. Fitting for
b
, the data indicate
n
th
eff
, spanning 0.71 to
4.5, with uncertainty all within

0
.
09
quanta.
To quantify the contributions due to quantum fluctua-
tions and classical cavity noise, we fix the cooling-tone
power at
n
cool
p
¼
4
×
10
5
(
γ
M
¼
2
π
×
360
Hz) and measure
the imbalance
n
m
n
þ
m
as we sweep
n
th
eff
. At each level, we
measure the average noise-power density
η
over a 250-Hz
window centered at
ω
c
and away from any motional
sideband. Over this range,
η
contains two contributions:
the noise radiating out of the microwave resonator, propor-
tional to
n
th
eff
, and the detector noise floor, set by the noise
temperature of the cryogenic amplifier (
T
N
3
.
6
K). We
directly measure the detector noise floor by switching from
the device to an impedance-matched bypass connection and
measure the noise-power density
η
0
over the same window
with matching detected tone powers.
In Fig.
3(d)
, we plot the sideband imbalance against the
noise floor increase
Δ
η
¼
η
η
0
, which is expected to
follow
n
m
n
þ
m
¼
2
n
th
eff
þ
1
¼
4
λ
Δ
η
þ
1
, where
λ
is the
conversion factor for
Δ
η
in units of cavity quanta
n
th
c
. The
data clearly follow a linear trend with a slope of
λ
¼ð
2
.
7

0
.
1
Þ
×
10
1
ð
aW
=
Hz
Þ
1
. More importantly,
we observe an offset of
1
.
2

0
.
2
, in excellent agreement
with the expected quantum imbalance of
þ
1
from the
quantum fluctuations of the microwave field.
As an additional check, we also consider the sideband
average
ð
n
þ
m
þ
n
m
Þ
=
2
as a function of
Δ
η
. Averaging
Eqs.
(3)
and
(4)
, we see that the resulting occupation
̄
n
m
þ
ð
β
=
2
Þ
does depend on
n
th
eff
due to the coupling between the
mechanical and microwave modes,
̄
n
m
¼ð
γ
m
=
γ
tot
Þ
n
th
m
þ
ð
γ
opt
=
γ
tot
Þð
2
n
th
c
þ
α
Þþð
γ
cool
opt
=
γ
tot
Þ
n
th
c
, where
γ
opt
(
γ
cool
opt
)is
the optical coupling rate for the individual probe (cooling)
n
th
eff
n
+
m
n
m
+
n
+
m
n
m
)/2
(
0
2
4
6
8
1
3
5
7
012345
(aW/Hz)
Δ
η
0
0.54
1.1
1.6
2.1
2.7
Microwave shot noise
n
+
m
n
+
m
n
m
/
10
12
8
6
4
1
2
3
4
5
0.71(5)
1.68(7)
2.30(7)
3.01(7)
4.54(9)
n
th
eff
-5.5
-4.5
4.5
5.5
65
45
aW/Hz
(kHz)
ω
(
ω
)
/
2
π
c
S
II
,
tot
ω
c
δ
+
ω
c
δ
ω
n
+
m
n
m
(a)
(b)
(c)
(d)
FIG. 3. Sideband asymmetry. (a) Pump scheme. Three tones are placed about the microwave resonance. Two probe tones generate up-
converted (red) and down-converted (blue) sidebands. An additional tone (green) cools the mechanical mode. (b) Sideband spectra.
̄
S
II;
tot
½
ω

measured at
n
th
eff
¼
0
.
60
(blue) and 2.5 (orange) with
̄
n
m
¼
5
.
3
and 7.1, respectively. (c) Sideband asymmetry. The ratio
n
m
=n
þ
m
versus
n
þ
m
is plotted for increasing noise injection. (d) Sideband imbalance (blue) and sideband average (red) versus the measured noise
increase
Δ
η
. Sideband imbalance
n
m
n
þ
m
and average
ð
n
m
þ
n
þ
m
Þ
=
2
exhibit a linear trend with
Δ
η
. The imbalance at
Δ
η
¼
0
is the
quantum imbalance due to the squashing of fluctuations of the microwave field.
A. J. WEINSTEIN
et al.
PHYS. REV. X
4,
041003 (2014)
041003-6
tones. Accounting for this so-called backaction heating of
the mechanical mode
[13,22]
, we recover
λ
¼ð
2
.
5

0
.
2
Þ
×
10
1
ð
aW
=
Hz
Þ
1
, consistent with the imbalance results
above.
Notably, the average sideband occupation does contain
contributions from mechanical zero-point fluctuations.
Future experiments could infer the mechanical quantum
contribution of
ð
β
=
2
Þ
with a method to independently
calibrate
̄
n
m
to high accuracy, for example, with a passively
cooled high-frequency mechanical mode thermalized to a
primary low-temperature thermometer.
In summary, we report the quantum imbalance between
the up- and down-converted motional sideband powers in a
cavity-electromechanical system measured with a symmet-
ric, linear detector. We show that for linear detection of the
microwave field, the imbalance arises from the correlations
between the mechanical motion and the quantum fluctua-
tions of the microwave field. For normal-ordered detection
of the microwave field, however, the imbalance arises
directly from the quantum fluctuations of the mechanics.
By further assuming that the output microwave field
satisfies the canonical commutator, which also determines
the quantum fluctuations of the mechanical mode, the
measurement can be interpreted as performing either
symmetric or normal-ordered detection regardless of the
type of detector utilized. In either interpretation, the
imbalance in motional sideband power stems from two
components: the quantum fluctuations of the internal fields
and the classical thermal noise from the environment. Once
the classical contribution is reduced or calibrated to well
below the quantum noise level, sideband imbalance pro-
vides a quantum-calibrated thermometer for mesoscopic
mechanical systems.
ACKNOWLEDGMENTS
We would like to acknowledge Yanbei Chen and
Matthew Woolley for helpful discussions. This work is
supported by funding provided by the Institute 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), by the NSF
(NSF-DMR 1052647 and NSF-EEC 0832819), and by the
DARPA ORCHID Program under a grant from AFOSR.
Note added.
Recently, two other groups have reported
similar asymmetry measurements
[25,26]
. Both groups
have used linear measurement schemes and are subject
to the same interpretation issues described here.
APPENDIX A: INPUT-OUTPUT THEORY
In this section, we give a framework to calculate the
output-noise spectrum of an opto- or electromechanical
system with arbitrary pump configuration by utilizing the
input-output theory. As a first example, we analyze an ideal
(without intrinsic losses) two-port opto- or electromechani-
cal system with a single pump tone either at frequency
ω
p
¼
ω
c
ω
m
or
ω
p
¼
ω
c
þ
ω
m
and discuss the origin of
the sideband asymmetry in the output-noise spectrum. We
then use this method to study the system in our experiment,
i.e., a two-port electromechanical system with three pumps
(two balanced detuned tones and a cooling tone).
We start with the standard Hamiltonian of an opto- or
electromechanical system
ˆ
H
¼
ω
c
ˆ
a
ˆ
a
þ
ω
m
ˆ
b
ˆ
b
þ
g
0
ˆ
a
ˆ
a
ð
ˆ
b
þ
ˆ
b
Þ
þ
ˆ
H
drive
þ
ˆ
H
diss
;
ð
A1
Þ
where
ˆ
a
(
ˆ
a
) is the annihilation (creation) operator of the
cavity field.
ˆ
b
(
ˆ
b
) is the annihilation (creation) operator of
the phonon, and
g
0
is the coupling strength between the
cavity and the mechanical oscillator. We assume an external
driving, described by
ˆ
H
drive
, which is applied on the input
port on the left side of the cavity. The optical and the
mechanical systems are both coupled to dissipative baths,
described by
ˆ
H
diss
, giving rise to the decay rates
γ
m
for the
mechanical and
κ
for the optical systems. The total cavity
linewidth
κ
consists of the contributions from the different
decay channels, namely, the right (
R
) and the left (
L
) ports,
as well as from intrinsic losses (
I
) inside the cavity,
i.e.,
κ
¼
κ
R
þ
κ
L
þ
κ
I
.
For large pumping fields, we may split the fields into
classical and quantum components
ˆ
a
̄
a
þ
ˆ
d
and
ˆ
b
̄
b
þ
ˆ
c
, where
ˆ
d
and
ˆ
c
describe the quantum fluctua-
tions of the cavity photon and the phonon. By using
input-output theory and neglecting the second-order con-
tributions from the quantum fluctuations, the linearized
quantum Langevin equations are
_
ˆ
d
¼
i
ω
0
c
ˆ
d
κ
2
ˆ
d
ig
0
̄
a
ð
t
Þ½
ˆ
c
þ
ˆ
c

X
σ
L;R;I
ffiffiffiffiffi
κ
σ
p
ˆ
d
σ
;
in
;
ð
A2a
Þ
_
ˆ
c
¼
i
ω
m
ˆ
c
γ
m
2
ˆ
c
ig
0
½
̄
a

ð
t
Þ
ˆ
d
þ
̄
a
ð
t
Þ
ˆ
d

ffiffiffiffiffi
γ
m
p
ˆ
c
in
;
ð
A2b
Þ
where
ω
0
c
¼
ω
c
þ
g
ð
̄
b
þ
̄
b

Þ
ω
c
. Including the possibil-
ity of multiple drives at frequencies
ω
n
, we obtain
̄
a
ð
t
Þ¼
P
n
̄
a
n
e
i
ω
n
t
as the driving field inside the cavity, with
̄
a
n
¼
ffiffiffiffiffi
κ
L
p
α
n
=
½ð
κ
=
2
Þ
i
ð
ω
n
ω
c
Þ
. Without loss of gen-
erality, we take
̄
a
n
to be real. In Eqs.
(A2)
,
ˆ
d
σ
;
in
describes
the input fluctuations to the cavity from channel
σ
with
damping rate
κ
σ
, and
ˆ
c
in
describes the input fluctuations to
the mechanical oscillator. The input field operators satisfy
the following commutation relations:
OBSERVATION AND INTERPRETATION OF MOTIONAL
...
PHYS. REV. X
4,
041003 (2014)
041003-7