of 24
Observation and interpretation of motional sideband asymmetry in a
quantum electro-mechanical 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
& 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, McGill University, Montreal, QC, H3A 2T8 CA
Quantum electro-mechanical systems offer a unique opportunity to probe quantum noise properties in macro-
scopic devices, properties which ultimately stem from the Heisenberg Uncertainty Principle. A simple example
of this 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. Due to quantum vac-
uum 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.
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 properites of both the system and the measurement scheme
1
. An
excellent illustration is found when considering measurements of the quantum harmonic oscillator. If measured with
an ideal energy detector, the observed signals will demonstrate energy level quantization
2, 3
; measured instead with an
ideal position detector, no evidence of quantized energy levels are found and the measured signals appear to be that
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
(
ω
)
6
=
S
xx
(+
ω
)
, where
S
xx
(
ω
)
is the spectral density of the observable
x
(
t
)
, defined as the Fourier transform of
ˆ
x
(
t
x
(0)
7
. For a quantum harmonic
oscillator, the negative and positive frequency sides of this spectral density describes 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
(
ω
m
) = 0
.
It can, however, absorb energy and as a result,
S
xx
(+
ω
m
) =
4
γ
m
x
2
zp
, where
x
zp
=
̄
h/
2
m
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 oscillator in a thermal state with occupation factor,
̄
n
m
, the spectral densities follow
S
xx
(
ω
m
) =
4
γ
m
x
2
zp
̄
n
m
and
S
xx
(+
ω
m
) =
4
γ
m
x
2
zp
( ̄
n
m
+ 1)
.
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 was detected via photodetection.
Analogous quantum noise effects can also be studied in macroscopic mechanical systems, using electro-mechanical
and opto-mechanical devices prepared and probed at quantum limits
5, 11–13
. These systems exhibit the Raman-like pro-
cesses of 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 asymmetry in the
mechanical quantum noise spectral density
S
xx
(
±
ω
m
)
. Recent experiments in optomechanics have demonstrated this
1
arXiv:1404.3242v1 [quant-ph] 11 Apr 2014
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) which probe the mechanical motion.
We also address a subtlety about these measurements which originates from their use of linear detection of the
scattered electromagnetic field: they measure the field amplitude (e.g. via heterodyne detection). This is contrast to
measurements based employing direct photodetection, where one filters the output light and counts photons associated
with a motional sideband. Although the predicted and measured motional sideband asymmetry 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 SI). 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 electromagnetic field fluctuations (classical and quantum); the intrinsic
quantum fluctuations of the mechanical mode contribute equally to the up and down-converted spectrum. In contrast,
in experiments which employ direct photodetection
8
, the imbalance in the output spectrum (in the absence of thermal
electromagnetic noise) is naturally attributed to asymmetric quantum noise of the mechanical motion. After a brief
discussion of these theoretical issues, we present measurements of the imbalance in a microwave-frequency electro-
mechanical device.
Theory
We begin with the Hamiltonian of our electro-mechanical system,
ˆ
H
= ̄
c
ˆ
a
ˆ
a
+ ̄
m
ˆ
b
ˆ
b
+ ̄
hg
0
ˆ
a
ˆ
a
(
ˆ
b
+
ˆ
b
)
(1)
where
ˆ
a
(
ˆ
a
)
is the annihilation (creation) operator of the microwave resonator mode with frequency
ω
c
,
ˆ
b
(
ˆ
b
)
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 clasical and quantum parts,
ˆ
a
=
e
p
t
( ̄
a
+
ˆ
d
)
and
ˆ
b
=
̄
b
+ ˆ
c
,
we linearize to obtain the following Heisenberg-Langevin equations
̇
ˆ
d
=
(
i
∆ +
κ
2
)
ˆ
d
iG
(
ˆ
c
+ ˆ
c
)
σ
=
R,L
κ
σ
ˆ
d
σ,
in
,
̇
ˆ
c
=
(
m
+
γ
m
2
)
ˆ
c
iG
(
ˆ
d
+
ˆ
d
)
γ
m
ˆ
c
in
,
where
∆ =
ω
c
ω
p
,
G
=
g
0
|
̄
a
|
, and
κ
=
κ
L
+
κ
R
(
γ
m
) is the microwave (mechanical) resonator damping rate.
The operators
ˆ
d
σ,
in
(
t
)
,
ˆ
c
in
(
t
)
describe noise incident on the microwave and mechanical resonator, respectively, and
satisfy:
ˆ
d
σ,
in
(
t
)
ˆ
d
σ
,
in
(
t
)
=
(
n
th
σ
+
α
)
δ
σ,σ
δ
(
t
t
)
,
ˆ
d
σ,
in
(
t
)
ˆ
d
σ
,
in
(
t
)
=
n
th
σ
δ
σ,σ
δ
(
t
t
)
,
ˆ
c
in
(
t
) ˆ
c
in
(
t
)
=
(
n
th
m
+
β
)
δ
(
t
t
)
,
ˆ
c
in
(
t
) ˆ
c
in
(
t
)
=
n
th
m
δ
(
t
t
)
,
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
,
β
describe the quantum vacuum fluctuations driving the microwave and mechanical resonators,
2
respectively; we have
α
=
β
= 1
, consistent with the uncertainty principle and the canonical commutation relation 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 further specialize to the case where a single microwave cavity drive is applied at
ω
p
=
ω
c
with
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
4
G
2

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.
For amplitude detection either with a linear amplifier as in this experiment, or optical heterodyne detection
14, 15
,
the symmetric noise spectrum is:
̄
S
II,
tot
[
ω
] =
1
2
dt
ˆ
I
tot
(
t
)
ˆ
I
tot
(0) +
ˆ
I
tot
(0)
ˆ
I
tot
(
t
)
e
iωt
.
(2)
with the amplitude of the output field
ˆ
I
tot
=
ˆ
d
R,
out
+
ˆ
d
R,
out
and where
ˆ
d
R,
out
=
ˆ
d
R,
in
+
κ
R
ˆ
d
. The output spectrum
near the cavity resonance for the two choices of drive detuning are found to be
̄
S
II,
tot
[
ω
]
∆=+
ω
m
=
̄
S
0
+
κ
R
κ
γ
opt
γ
m
(
γ
m
2
)
2
+ (
ω
ω
c
)
2
[(
n
th
m
+
β
2
)
(
n
th
eff
+
α
2
)
]
,
(3)
̄
S
II,
tot
[
ω
]
∆=
ω
m
=
̄
S
0
+
κ
R
κ
γ
opt
γ
m
(
γ
m
2
)
2
+ (
ω
ω
c
)
2
[(
n
th
m
+
β
2
)
+
(
n
th
eff
+
α
2
)
]
.
(4)
where for
∆ =
ω
m
(
∆ =
ω
m
), the up- (down-) converted sideband is centered on the cavity resonance. The
noise floor for both cases is given by
̄
S
0
=
α/
2 +
n
th
R
+ 4
κ
R
(
n
th
c
n
th
R
)
, and we have defined
n
th
eff
= 2
n
th
c
n
th
R
(where
n
th
c
= (
κ
L
n
th
L
+
κ
R
n
th
R
)
is the effective cavity thermal occupancy). In Fig.1(c), we illustrate the underlying
components of this spectrum.
One sees explicitly that the sideband imbalance,
̄
S
II,
tot
[
ω
]
|
∆=
ω
m
̄
S
II,
tot
[
ω
]
|
∆=+
ω
m
, is proportional to
(2
n
th
eff
+
α
)
, and hence is entirely due to fluctuations in the microwave fields driving the cavity. This is true both when this
noise is thermal, and when it is purely quantum (i.e.
n
th
R
=
n
th
L
= 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 SI for further insights based on a scattering approach). This is the basic mechanism
of noise squashing, which in the case of thermal noise was previously observed in a cavity electromechanical system
4
. This mechanism can also be fully described using a general linear measurement formalism
16
, where it is attributed
to the presence of correlations between the backaction and imprecision noises of the detector, correlations which are
out-of-phase and have magnitude
̄
h/
2
in the zero-temperature limit. Interestingly, this precise value plays a special
role in the theory of quantum limits on linear amplification
7
(see SI 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
+
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 which almost entirely
determines 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.
3
In contrast, direct measurement of the sideband signal via photon counting yields the normal ordered spectrum,
S
N
II,
tot
(
ω
) =
dt
:
ˆ
I
tot
(
t
)
ˆ
I
tot
(0) :
e
iωt
,
(5)
with output spectra given by
S
N
II,
tot
[
ω
]
∆=+
ω
m
=
(
̄
S
0
α
2
)
+
κ
R
κ
γ
opt
γ
m
(
γ
m
2
)
2
+ (
ω
ω
c
)
2
(
n
th
m
n
th
eff
)
,
(6)
S
N
II,
tot
[
ω
]
∆=
ω
m
=
(
̄
S
0
α
2
)
+
κ
R
κ
γ
opt
γ
m
(
γ
m
2
)
2
+ (
ω
ω
c
)
2
(
n
th
m
+
β
+
n
th
eff
)
.
(7)
Note that when one sets
α
=
β
= 1
, the asymmetry of these normal-ordered spectra,
S
N
II,
tot
[
ω
]
|
∆=
ω
m
S
N
II,
tot
[
ω
]
|
∆=+
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 direct photodetection, one cannot
attribute the zero-temperature sideband asymmetry to a correlation between backaction-driven position fluctuations
and imprecision noise, as there is no imprecision noise floor.
While the above simple calculations suggest that the sideband asymmetry 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,
[
ˆ
d
R,
out
[
ω
]
,
ˆ
d
R,
out
[
ω
]
]
=
αδ
(
ω
+
ω
)
. It necessarily follows that the spectra in Eqs. (2) and Eqs. (5)
will differ only by a frequency-independent noise floor of magnitude
α/
2
16
. 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
involves an additional assumption on the value of
β
: (if
β
6
=
α
, then the output commutator would not be the same as
the input, see SI).
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.
Experiment
Our system is composed of a superconducting microwave resonator, also referred to as “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
= 2
π
×
5
.
4
GHz, total loss rate
κ
= 2
π
×
860
kHz,
output coupling rate
κ
R
= 2
π
×
450
kHz, and input coupling rate
κ
L
= 2
π
×
150
kHz. The capacitor top gate is a
flexible aluminum membrane (40
μ
m
×
40
μ
m
×
150nm) with a fundamental drumhead mode with resonance frequency
ω
m
= 2
π
×
4
.
0
MHz and intrinsic loss rate
γ
m
= 2
π
×
10
Hz at 20mK. Motional displacement of the top gate
modulates the microwave resonance frequency with an estimated coupling rate of
g
0
=
∂ω
c
∂x
x
zp
= 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 at 300K; cryogenic attenuators further reduce the noise down
to the shot noise level
4
. A pair of microwave switches at the device stage select between the device or a bypass
connection for high precision noise floor calibration of the cryogenic amplifier. The output signal passes through two
cryo-circulators at
100mK followed by a cryogenic low-noise amplifier at 4.2K, and finally to a 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 50mK, 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
4
cavity at zero pumping and comparing this to power spectra with the bypass switched in place (see SI), we conclude
that there is a small contribution to
n
th
c
due to thermal radiation from the isolated port of the cryogenic circulators,
given by the occupation factor
n
th
R
= 0
.
34
±
0
.
03
.
When a single microwave tone is applied to the device at
ω
p
, the parametric coupling converts mechanical os-
cillations 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 microwave
resonance suppresses motional sidebands outside of the linewidth and we consider only the contributions of signals
converted to frequencies near
ω
c
. These are the Lorentzian components of the noise power spectra of Eqs. (3) and (4),
which for the remainder of the paper are denoted by “+” and “-”, respectively, and are labeled in Fig.1(c).
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 back-action cooling
18
.
Two additional probe tones, placed at
ω
c
±
(
ω
m
+
δ
)
, produce up and down converted sidebands symmetrically
detuned from cavity center (Fig.3(a)). The detunings are chosen to ensure no interference between the sidebands
(
δ
c
= 2
π
×
30
kHz,
δ
= 2
π
×
5
kHz) so that we may consider the probe sidebands as independent measurements of
the dressed mechanical mode as validated by theory.
To summarize the main differences between the simplified theory model presented above and our actual ex-
periment, 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 pow-
ers. From further analysis (see SI), we estimate corrections to the sideband asymmetry that are

1
and far below the
measurement resolution of our system.
To convert the motional sideband powers into equivalent mechanical occupation, we turn off the cooling tone and
measure the probe sidebands (
δ
= 2
π
×
500
Hz) with low optical damping (
n
+
p
=
n
p
'
5
×
10
2
) and high mechanical
occupation set by the cryostat temperature. Regulating the temperature to calibrated levels between 20 to 200mK,
we calculate the integrated noise 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
.
19
As we vary the cryostat temperature,
T
, we compare the normalized power to the
thermal occupation factor
[exp(
̄
m
k
B
T
)
1]
1
(Fig.2(b)). A linear fit yields the conversion factors for the up-converted
(
n
+
m
) and down-converted (
n
m
) sidebands:
n
+
m
= (9
.
9
±
0
.
2)
×
10
8
·
P
+
m
/P
+
thru
and
n
m
= (5
.
4
±
0
.
1)
×
10
8
·
P
m
/P
thru
.
Further detuning the probe tones (
δ
= 2
π
×
5
kHz) and turning on the cooling tone (
δ
c
= 2
π
×
30
kHz), we
explore the sideband ratio,
n
m
/n
+
m
, over various the mechanical and microwave occupations. To reduce
̄
n
m
to values
approaching 1, we increase the cooling tone power up to
n
cool
p
= 4
×
10
5
. For sideband characterization, the probe
tone powers are set to
n
p
=
n
+
p
= 10
5
and the probe sideband spectra are analyzed using the conversion factors
described above. The imbalance between
n
m
and
n
+
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
n
m
/n
+
m
as a function of
n
+
m
in Fig.
3(c). Each curve corresponds to one setting of injected microwave noise. The data shows excellent agreement to the
expected ratio,
n
m
/n
+
m
= 1 + (2
n
th
eff
+ 1)
/n
+
m
. This relationship highlights the combined effect of quantum and
classical noise in Eqs. (3) and (4) (see SI). By fitting each curve to a two parameter model:
a
+
b/n
+
m
, we find an
average constant offset
a
= 0
.
99
±
0
.
02
for all curves, accurately matching the model and confirming our calibration
techniques. Fitting for
b
, the data indicates
n
th
eff
spanning
0
.
71
to
4
.
5
with uncertainty all within
±
0
.
09
quanta.
To quantify the contributions due to quantum fluctuations and classical cavity noise, we fix the cooling tone
power at
n
cool
p
= 4
×
10
5
(
γ
M
= 2
π
×
360
Hz) and measure the imbalance
n
m
n
+
m
as we sweep
n
th
eff
. At each
5
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,
proportional 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,
η
=
η
η
0
, which is expected
to follow:
n
m
n
+
m
= 2
n
th
eff
+ 1 = 4
λ
·
η
+ 1
, where
λ
is the conversion factor for
η
in units of cavity quanta,
n
th
c
. The data clearly follows a linear trend with a slope of
λ
= (2
.
7
±
0
.
1)
×
10
1
(aW/Hz)
1
. More importantly, we
observe an offset of
1
.
2
±
0
.
2
, in excellent agreement with the expected quantum imbalance of “+1” from the quantum
fluctuations of the microwave field.
As an additional check, we also consider the sideband average,
(
n
+
m
+
n
m
)
/
2
, as a function of
η
. Averaging
Eqs. (3) and (4), we see that the resulting occupation,
̄
n
m
+
β
2
, does depend on
n
th
eff
due to the coupling between the
mechanical and microwave modes,
̄
n
m
=
γ
m
γ
tot
n
th
m
+
γ
opt
γ
tot
(2
n
th
c
+
α
)+
γ
cool
opt
γ
tot
n
th
c
, where
γ
opt
(
γ
cool
opt
) is the optical coupling
rate for the individual probe (cooling) tones. Accounting for this so-called back-action heating of the mechanical
mode
13, 18
, we recover
λ
= (2
.
5
±
0
.
2)
×
10
1
(aW/Hz)
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
β
2
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 electro-mechanical system measured with a symmetric, 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
fluctuations of the microwave detection field. For normal-ordered detection of the microwave field, however, the im-
balance arises directly from the quantum fluctuations of the mechanics. By further assuming that the output microwave
field satisfies the cannonical 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 both scenarios, the imbalance in motional sidebands is a fundamental quantity originating from
the Heisenberg’s uncertainty relations and provides a quantum calibrated thermometer for mesoscopic mechanical
systems.
1. Braginsky, V. B. & Khalili, F. Y.
Quantum Measurement
(Cambridge University Press, 1992).
2. Santamore, D. H., Doherty, A. C. & Cross, M. C. Quantum nondemolition measurement of fock states of meso-
scopic mechanical oscillators.
Phys. Rev. B
70
, 144301 (2004).
3. Clerk, A. A., Marquardt, F. & Harris, J. G. E. Quantum measurement of phonon shot noise.
Phys. Rev. Lett.
104
,
213603 (2010).
4. Roucheleau, T.
et al.
Preparation and detection of a mechanical resonator near the ground state of motion.
Nature
463
, 72–75 (2010).
5. Teufel, J.
et al.
Sideband cooling of micromechanical motion to the quantum ground state.
Nature
475
, 359–363
(2011).
6. Muthukrishnan, A., Scully, M. O. & Zubairy, M. S.
The nature of light: what is a photon?
(CRC Press, 2008).
6
7. Clerk, A. A., Devoret, M. H., Girvin, S. M., Marquardt, F. & Schoelkopf, R. J. Introduction to quantum noise,
measurement, and amplification.
Rev. Mod. Phys.
82
, 1155–1208 (2010).
8. Diedrich, F., Bergquist, J., Itano, W. & Wineland, D. Laser cooling to the zero-point energy of motion.
Phys. Rev.
Lett.
62
, 403 (1989).
9. Jessen, P.
et al.
Observation of quantized motion of rb atoms in an optical field.
Physical review letters
69
, 49
(1992).
10. Monroe, C.
et al.
Resolved-sideband raman cooling of a bound atom to the 3d zero-point energy.
Physical Review
Letters
75
, 4011 (1995).
11. Chan, J.
et al.
Laser cooling of a nanomechanical oscillator into its quantum ground state.
Nature
478
, 89–92
(2011).
12. Purdy, T., Peterson, R. & Regal, C. Observation of radiation pressure shot noise on a macroscopic object.
Science
339
, 801–804 (2013).
13. Suh, J.
et al.
Mechanically detecting and avoiding the quantum fluctuations of a microwave field.
arXiv:1312.4084
(2013).
14. Safavi-Naeini, A. H.
et al.
Observation of quantum motion of a nanomechanical resonator.
Phys. Rev. Lett.
108
,
033602 (2012).
15. Brahms, N., Botter, T., Schreppler, S., Brooks, D. W. & Stamper-Kurn, D. M. Optical detection of the quantization
of collective atomic motion.
Physical Review Letters
108
, 133601 (2012).
16. Khalili, F. Y.
et al.
Quantum back-action in measurements of zero-point mechanical oscillations.
Phys. Rev. A
86
,
033840 (2012).
17. Walls, D. F. & Milburn, G.
Quantum optics
(Springer-Verlag, 1995).
18. Marquardt, F., Chen, J. P., Clerk, A. A. & Girvin, S. M. Quantum theory of cavity-assisted sideband cooling of
mechanical motion.
Phys. Rev. Lett.
99
, 093902 (2007).
19. Hertzberg, J. B.
et al.
Backaction evading measurements of nanomechanical motion.
Nature Physics
463
, 72–75
(2009).
Acknowledgements
We would like to acknowledge Yanbei Chen and Matthew Woolley for helpful discussion.
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 NSF (nsf-dmr 1052647, nsf-eec 0832819), and by the DARPA ORCHID pro-
gram under a grant from AFOSR.
Competing Interests
The authors declare no competing financial interests.
Correspondence
Correspondence and requests for material should be addressed to Keith Schwab (email: schwab@caltech.edu).
7
Figure 1:
Comparison between photodetection and linear detection.
a.
Pump scheme. We consider a single mi-
crowave cavity (dotted line) pumped at
ω
c
±
(
ω
m
+
δ
)
(green). The up-converted (red) and down-converted (blue)
motional sidebands are placed tightly within the cavity linewidth. For figure clarity, the occupation of the microwave
and mechanical modes are assumed to be zero.
b.
Normal-ordered detection. Photodetection is sensitive to the asym-
metric motional noise spectrum,
S
xx
. The photodetector is not sensitive to microwave shot noise and the noise floor
(
S
II
) is from detector non-idealities (light grey), analogous to dark counts for a photodetector.
c.
Linear detection.
The contribution from the symmetrized motional noise,
̄
S
xx
, is present in both sidebands. Microwave shot noise (dark
grey) and amplifier noise (light grey) 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 sqashing (red)
and anti-squashing (blue) of the noise floor. Though the source is different, the sideband imbalance is identical in both
photodetection and linear detection. For mathematical description of
S
II
and
S
IF
, refer to SI.
8
Figure 2:
Device, calibration, and measurement scheme.
a.
Electron micrograph of the measured device. A sus-
pended aluminum (grey) membrane patterned on silicon (blue) forms the electro-mechanical 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 co-planar waveguides.
b.
Motional sideband calibration. The cryostat
temperature is regulated while the mechanical mode is weakly probed with microwave tones set at
ω
c
+
ω
m
+
δ
(blue)
and at
ω
c
ω
m
δ
(red) detunings, with
δ
= 2
π
×
500
Hz. The observed linear dependence provides the calibration
between the normalized sideband power and the mechanical occupation factor. Inset, up-converted motional sideband
spectra collected at 20mK (top) and 200mK (bottom), with
ω
=
ω
(
ω
c
δ
)
.
c.
Schematic of the microwave
measurement circuit.
9
Figure 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 (purple) cools the
mechanical mode.
b.
Sideband spectra.
̄
S
II,
tot
(
ω
)
measured at
n
th
eff
= 0
.
60
(blue) and
2
.
5
(purple) with
̄
n
m
=
4
.
7
±
0
.
1
.
c.
Sideband asymmetry. The ratio
n
m
/n
+
m
vs.
n
+
m
is plotted for increasing noise injection.
d.
Sideband
imbalance (blue) and sideband average (purple) vs. the measured noise increase,
η
. Sideband imbalance,
n
m
n
+
m
,
and average,
(
n
m
+
n
+
m
)
/
2
, exhibit a linear trend with
η
. The imbalance at
η
= 0
is the quantum imbalance due
to the squashing of fluctuations of the microwave field.
10