of 10
PHYSICAL REVIEW A
86
, 033840 (2012)
Quantum back-action in measurements of zero-point mechanical oscillations
Farid Ya. Khalili,
1
Haixing Miao,
2
Huan Yang,
2
Amir H. Safavi-Naeini,
3
Oskar Painter,
3
and Yanbei Chen
2
1
Physics Faculty, Moscow State University, Moscow 119991, Russia
2
Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, California 91125, USA
3
Thomas J. Watson, Sr. Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 8 June 2012; published 25 September 2012)
Measurement-induced back-action, a direct consequence of the Heisenberg uncertainty principle, is the defining
feature of quantum measurements. We use quantum measurement theory to analyze the recent experiment of
Safavi-Naeini
et al.
[
Phys.Rev.Lett.
108
, 033602 (2012)
], and show that the results of this experiment not only
characterize the zero-point fluctuation of a near-ground-state nanomechanical oscillator, but also demonstrate the
existence of quantum back-action noise—through correlations that exist between sensing noise and back-action
noise. These correlations arise from the quantum coherence between the mechanical oscillator and the measuring
device, which build up during the measurement process, and are key to improving sensitivities beyond the
standard quantum limit.
DOI:
10.1103/PhysRevA.86.033840
PACS number(s): 42
.
50
.
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.
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.
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.
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.
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.
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I. INTRODUCTION
Quantum mechanics dictates that no matter or field can stay
absolutely at rest, even at the ground state, for which energy
is at minimum. A starting point for deducing this inevitable
fluctuation is to write down the Heisenberg uncertainty
principle
[
ˆ
x,
ˆ
p
]
=
i
̄
h,
(1)
which leads to
xp

̄
h/
2
.
(2)
Here
ˆ
x
and
ˆ
p
are the position and momentum operators,
while
x
and
p
are the standard deviations of position
and momentum for an arbitrary quantum state. Equation
(2)
means that we cannot specify the position and momentum of a
harmonic oscillator simultaneously as a point in classical phase
space—the oscillator must at least occupy ̄
h/
2 area in the phase
space. If the oscillator has mass of
m
and eigenfrequency of
ω
m
, then in the Heisenberg picture we can write
[
ˆ
x
q
(
t
)
,
ˆ
x
q
(
t

)]
=
i
̄
h
sin
ω
m
(
t

t
)
m
,
(3)
which leads to
x
q
(
t
)
x
q
(
t

)

̄
h
|
sin
ω
m
(
t

t
)
|
2
m
,
(4)
with
ˆ
x
q
(
t
) being the Heisenberg operator of the oscillator
position, quantum mechanically evolving under the free
Hamiltonian. Here
x
q
(
t
) is the standard deviation of
ˆ
x
q
(
t
)
for an arbitrary quantum state. Equation
(4)
means that the
position of a freely evolving quantum harmonic oscillator
cannot continuously assume precise values, but instead must
fluctuate. This fluctuation carries the zero-point mechanical
energy of ̄
m
/
2.
As a key feature of quantum mechanics, the zero-point
fluctuation of displacement is an important effect to verify
when we bring macroscopic mechanical degrees of freedom
into their ground states [
1
8
]. Needless to say, a continuous
observation of the zero-point fluctuation of a macroscopic
mechanical oscillator requires superb displacement sensitivity.
However, what constitutes an “observation of the quantum
zero-point fluctuation” is conceptually subtle. Equations
(3)
and
(4)
, which argue for the inevitability of the zero-point
fluctuation, also dictate that the “exact amount” of the
zero-point fluctuation cannot be determined precisely. More
specifically, if we use a linear measurement device to probe
the zero-point fluctuation, which has an output field of
ˆ
y
(
t
),
then we must at least have
[
ˆ
y
(
t
)
,
ˆ
y
(
t

)]
=
0(5)
at all times in order for
ˆ
y
(
t
) to be able to represent an
experimental data string—with measurement noise simply
due to the projection of the device’s quantum state into
simultaneous eigenstates of all
{
ˆ
y
(
t
):
t
R
}
. This means
ˆ
y
must be written as
ˆ
y
(
t
)
=
ˆ

(
t
)
+
ˆ
x
q
(
t
)
,
(6)
with nonvanishing additional noise (error) ˆ

(
t
), which consists
of degrees of freedom of the measurement device and
compensates the nonvanishing commutator of
ˆ
x
q
.
1
In addition,
during the measurement process, the actual evolution of the
mechanical displacement
ˆ
x
must differ from its free evolution
ˆ
x
q
. This is because
[
ˆ
x
(
t
)
,
ˆ
x
(
t

)]
i
̄
h
χ
(
t

t
)(7)
is also the classical response function of
x
to an external
force: any device that attempts to measure
ˆ
x
by coupling
it with an external observable
ˆ
F
, which introduces a term
proportional to
ˆ
x
ˆ
F
into the Hamiltonian, will have to cause
nonzero disturbance. For this reason, we can expand the
measurement error ˆ

into two parts, i.e.,
ˆ
z
is the sensing
1
We note that Ozawa has developed a different formalism to
quantify the issues that arise when attempts are made to measure
noncommuting observables such as
ˆ
x
q
(
t
)[
9
,
10
]. However, we have
chosen to adopt the Braginsky-Khalili approach [
11
] because it is
immediately applicable when the noncommuting observable is acting
as a probe for an external classical force.
033840-1
1050-2947/2012/86(3)/033840(10)
©2012 American Physical Society
FARID YA. KHALILI
et al.
PHYSICAL REVIEW A
86
, 033840 (2012)
noise that is independent from mechanical motion and
ˆ
x
BA
is additional disturbance to the mechanical motion from the
measurement-induced back-action, and rewrite
ˆ
y
(
t
)as
ˆ
y
(
t
)
=
ˆ
z
(
t
)
+
ˆ
x
BA
(
t
)
︷︷
ˆ

(
t
)
+
ˆ
x
q
(
t
)
=
ˆ
z
(
t
)
+
ˆ
x
(
t
)
.
(8)
The mechanical displacement under measurement is therefore
a sum of the freely evolving operator
ˆ
x
q
plus the disturbance
ˆ
x
BA
due to back-action noise, namely,
ˆ
x
(
t
)
=
ˆ
x
q
(
t
)
+
ˆ
x
BA
(
t
).
The above lines of reasoning lie very much at the heart of
linear quantum measurement theory, pioneered by Braginsky
in the late 1960s, aiming to describe resonant-bar gravitational-
wave detectors [
11
,
12
] and later adapted to the analysis of laser
interferometer gravitational-wave detectors by Caves [
13
]. A
key concept in linear quantum measurement theory is the
trade-off between sensing noise and back-action noise, which
gives rise to the so-called standard quantum limit (SQL).
For optomechanical devices, sensing noise takes the form of
quantum shot noise due to the discreteness of photons, while
the quantum back-action is enforced by quantum fluctuations
in the radiation pressure acting on the mechanical oscillators
[
13
], which is therefore also called quantum radiation-pressure
noise. It has been shown that the SQL, although not a strict limit
for sensitivity, can only be surpassed by carefully designed
linear measurement devices, which take advantage of quantum
correlations between the sensing noise and the back-action
noise.
Observing signatures of quantum back-action, achieving
and surpassing the associated SQL in mechanical systems
are of great importance for the future of quantum-limited
metrology, e.g., gravitational-wave detections [
14
22
]. At
the moment, it is still experimentally challenging to directly
observe quantum radiation-pressure noise in optomechanical
devices due to high levels of environmental thermal fluctu-
ations, and there are significant efforts being made toward
this [
4
8
,
23
]. One approach proposed by Verlot
et al.
[
4
]is,
instead, to probe the quantum correlation between the shot
noise and the radiation-pressure noise, which, in principle, is
totally immune to thermal fluctuations.
In this paper, we analyze a recent experiment performed by
Safavi-Naeini
et al.
[
24
] in which a radiation-pressure-cooled
nanomechanical oscillator—the movable mirror of a high-
finesse cavity—is probed by a second beam of light, detuned
from the cavity, for its zero-point mechanical oscillation.
The output power spectrum of the second beam, near the
mechanical resonant frequency, serves as an indicator of the
oscillator’s zero-point motion. It was experimentally observed
that when the second beam is detuned on opposite sides
from the cavity resonance, the output power spectra turn
out to be different. By using the theory of linear quantum
measurements, we will show that this experiment not only
probes the zero-point fluctuation of the mechanical oscillator
at nearly ground state, but also illustrates vividly the nontrivial
correlations between sensing noise and back-action noise—a
much sought-after effect in the gravitational-wave-detection
community.
Its contribution to the output spectrum is equal
to the zero-point fluctuation for one detuning of the readout
beam, and exactly opposite for the other detuning.
The outline of this paper is as follows: In Sec.
II
, we will
give a brief overview of the experiment by Safavi-Naeini
et al.
and present an analysis of this experiment using quantum
measurement theory. In Sec.
III
, we will more broadly discuss
the nature of the mechanical zero-point fluctuation, and show
that in attempts to measure the zero-point fluctuation, the
contributions from sensing–back-action noise correlations can
generically be comparable to the zero-point fluctuation itself.
In addition, we will discuss linear quantum measurement
devices that use a near-ground-state mechanical oscillator as a
probe for external classical forces near its resonant frequency,
and show the limitation on the measurement sensitivity
imposed by the zero-point fluctuation and the connection to
the SQL. We will conclude in Sec.
IV
.
II. A TWO-BEAM EXPERIMENT THAT MEASURES
ZERO-POINT MECHANICAL OSCILLATION
We describe in Sec.
II A
the experiment performed by
Safavi-Naeini
et al.
, put its results into the framework of
linear quantum measurement theory in Sec.
II B
, and provide a
detailed analysis in Sec.
II C
. In Sec.
II D
, we comment on the
connection between the viewpoint from quantum measurement
and the scattering picture presented in Ref. [
24
].
A. Experimental setup and results
In the experiment, two spatial optical modes are coupled
to a mechanical vibrational mode in a patterned silicon
nanobeam. One spatial mode—the cooling mode—is pumped
with a relatively high power at a “red” detuning (lower than
resonance), and is used to cool the mechanical mode via
radiation pressure damping [
2
]; the other cavity mode—the
readout mode—has a much lower power and is used for
probing the mechanical motion. The readout laser frequency
ω
lr
is detuned from the resonant frequency
ω
r
of the readout
mode by either
+
ω
m
or
ω
m
. The observed spectra of the
readout laser are asymmetric with respect to the detuning,

ω
r
ω
lr
. Specifically, in the positive-detuning case,

=
ω
m
, the spectrum has a smaller amplitude than that
in the negative-detuning case. The area
I
+
enclosed by the
spectrum in the positive-detuning case,
after subtracting out
the noise floor away from the mechanical resonant frequency
,
is proportional to the thermal occupation number

n

of the
mechanical oscillator, while in the negative-detuning case, the
enclosed area is
I
∝
n
+
1. Such asymmetry is illustrated
in Fig.
1
.InRef.[
24
], we introduced the following figure of
FIG. 1. (Color online) The observed spectra of the readout laser
in the positive-detuning case (left) and the negative-detuning case
(right).
033840-2
QUANTUM BACK-ACTION IN MEASUREMENTS OF ZERO-
...
PHYSICAL REVIEW A
86
, 033840 (2012)
merit to quantify the asymmetry:
η
I
I
+
1
=
1

n

.
(9)
We interpreted this asymmetry as arising from the quantized
motion of the mechanical oscillator. The asymmetry is thus
assigned to the difference between the phonon absorption
rate, proportional to

n

, and the emission rate, proportional
to

n
+
1. This is completely analogous to that used for
the calibration of the motional thermometry of ions (atoms)
trapped in electrical (optical) traps [
25
28
]. Additionally, these
scattering processes have an underlying physics similar to
the bulk nonlinear Raman-scattering processes used in the
spectroscopic analysis of crystals [
29
,
30
], where an ensemble
of vibrational degrees of freedom internal to the molecular
structure of the system interacts with incident light. Typically,
in these nonlinear optics experiments, photon counters are
used to keep track of the (anti-)Stokes photons. In contrast,
in our experiment, a heterodyne measurement scheme was
used to find the amplitude quadrature of the readout mode.
Interestingly, by choosing the detuning

ω
m
, and in the
resolved-sideband regime, spectra of the amplitude quadrature
are equal to the emission spectra of the (anti-)Stokes photons
plus a constant noise floor due to vacuum fluctuation of the
light—the shot noise. We will elaborate on this point in
Sec.
II D
and show explicitly such a connection. Intuitively,
one can view the cavity mode as an optical filter to selectively
measure the emission spectra—for

=
ω
m
, the anti-Stokes
process is significantly enhanced as the emitted photon is on
resonance with respect to the cavity mode, and one therefore
measures the spectrum for the anti-Stokes photons; while for

=−
ω
m
, the spectrum of the Stokes photon is measured.
B. Interpretation in terms of quantum measurement
Here we provide an alternative viewpoint to Ref. [
24
],
emphasizing the role of quantum back-action and its relation
to quantization of the mechanical oscillator. First of all, we
separate the experimental system into two parts. The first
part includes the cooling mode, the mechanical oscillator, and
the environmental thermal bath that the oscillator couples to
(the left and middle boxes in Fig.
2
), which can be viewed
as providing an
effective mechanical oscillator
nearly at the
ground state, but with a quality factor significantly lower than
the intrinsic quality factor of the mechanical mode. It is the
zero-point fluctuation of this effective oscillator that we shall
be probing. The second part of the system consists of the
readout mode (the box on the right of Fig.
2
), which couples
to the effective oscillator (the first part of the system) through
displacement
ˆ
x
alone. The second part provides us with an
output
ˆ
y
, which contains information about the zero-point
fluctuation of the effective mechanical oscillator.
FIG. 2. (Color online) The relation among different parts of the
optomechanical system in the experiment. The thermal heat bath and
the cooling mode together create an effective quantum heat bath for
the mechanical oscillator, which in turn couples to the readout mode.
1. The mechanical oscillator near ground state
Let us focus on the first part of the system (left two boxes
of Fig.
2
), namely, the effective mechanical oscillator (since
this will be a stand-alone subject of study in later discussions,
we shall often ignore the word “effective”). The environmental
heat bath and the cooling mode together form a
quantum
heat
bath with fluctuation close to the zero-point value. In the steady
state, the “free” mechanical displacement is determined by its
coupling to this bath (“free” means the absence of the readout
mode):
ˆ
x
q
(
t
)
=
t
−∞
χ
(
t
t

)
ˆ
F
q
(
t

)
dt

.
(10)
Here
χ
is the response function of the mechanical oscillator,
and for a high-quality-factor oscillator, we have
χ
(
t
t

)
=−
[
ˆ
x
(
t
)
,
ˆ
x
(
t

)]
i
̄
h
=
e
κ
m
|
t
t

|
/
2
sin
ω
m
(
t
t

)
m
.
(11)
Note that we have an additional decay factor compared
with Eq.
(3)
, which describes an
idealized
free oscillator.
The decay rate
κ
m
here is determined jointly by the intrinsic
decay rate of the mechanical mode, and the optomechanical
interaction between the mechanical mode and the cooling
mode. The force
ˆ
F
q
lumps together the fluctuating force
acting on the mechanical mode by the environmental heat bath
and the cooling mode. If the oscillator approaches the ground
state only after applying the cooling mode, then one can show
that
ˆ
F
q
is dominated by fluctuation of the cooling mode.
The above two equations state that for a realistic mechanical
oscillator with nonzero decay rate, its zero-point fluctuation in
the steady state can be viewed as driven by the quantum heat
bath surrounding it. We will return to this prominent feature
of linear quantum systems later in Sec.
III A
.
2. The quantum measurement process
Let us now move on to the second part of the system (right
box of Fig.
2
), in which the readout mode serves as a linear
position meter that measures the mechanical displacement.
We can rewrite the disturbance
ˆ
x
BA
in Eq.
(8)
in terms of
the back-action force
ˆ
F
BA
arising from the radiation-pressure
fluctuation of the readout mode, namely,
ˆ
x
BA
(
t
)
=
t
−∞
χ
(
t
t

)
ˆ
F
BA
(
t

)
dt

.
(12)
We have assumed that the readout mode does not modify the
dynamics of the oscillator, which is a good approximation for
the low pumping power used in the experiment. Written in the
frequency domain, the readout mode output
ˆ
y
[cf. Eq.
(8)
]is
ˆ
y
(
ω
)
=
ˆ
z
(
ω
)
+
χ
(
ω
)
ˆ
F
BA
(
ω
)
+
χ
(
ω
)
ˆ
F
q
(
ω
)
,
(13)
where
χ
(
ω
)
=−
1
m
(
ω
2
ω
2
m
+
m
ω
)
(14)
is the Fourier transform of

(
t
)
χ
(
t
), with

the Heaviside
function, i.e., the positive half of
χ
(
t
) [even though
χ
(
t
)exists
for both
t>
0 and
t<
0]. The spectral density
S
yy
(
ω
)of
ˆ
y
then reads
S
yy
=
S
zz
+
2Re[
χ
S
zF
]
+|
χ
|
2
S
BA
FF
+|
χ
|
2
S
q
FF
.
(15)
033840-3
FARID YA. KHALILI
et al.
PHYSICAL REVIEW A
86
, 033840 (2012)
Here these single-sided spectral densities are defined in a sym-
metrized way (see Appendix
A
), which guarantees bilinearity
for the cross spectrum and positivity for the self-spectrum.
S
zz
and
S
BA
FF
are the sensing-noise and back-action force noise
spectra, respectively;
S
zF
is the cross correlation between
ˆ
z
and
ˆ
F
BA
; and the force spectrum of the effective quantum heat
bath made up by the environmental heat bath and the cooling
mode is given by
S
q
FF
=
(4

n
+
2) ̄
hmκ
m
ω
m
,
(16)
where

n

is the thermal occupation number.
3. Asymmetry between spectra
Experimentally, it was observed that the output spectra
S
yy
for the two opposite detunings,

ω
m
, are different—
given the same thermal occupation number for the oscillator,
S
yy
(
ω
)
|

=−
ω
m
=
S
yy
(
ω
)
|

=
ω
m
.
(17)
As we will show in Sec.
II C
, when we flip the sign of the
detuning

of the readout beam, the only term in
S
yy
that
changes is
S
zF
, which is the correlation between the sensing
noise and the back-action noise. According to Eq.
(34)
,wehave
S
zF
(
ω
)
≈−
i
̄
h
ω

(18)
in the resolved-sideband regime with the cavity bandwidth
κ
r
ω
m
, which is the case in the experiment. The asymmetry
factor defined in Eq.
(9)
is given by
η
=
2
Re[
χ
(
S
zF
S
+
zF
)]
[
|
χ
|
2
S
q
FF
+
2Re(
χ
S
+
zF
)
]
=
1

n

.
(19)
Here
S
±
zF
is defined by
S
±
zF
S
zF
|

ω
m
, and in particular
around the mechanical resonant frequency
ω
m
, where
S
±
zF
contribute to the above integral,
S
±
zF
≈∓
i
̄
h.
(20)
The asymmetry, or effect of quantum correlation
S
zF
,ismost
prominent when the thermal occupation number approaches
zero. Indeed, if we focus on the quantum fluctuation by taking
the limit of

n
→
0, then we obtain
2Re
(
χ
S
±
zF
)
=∓
|
χ
|
2
S
q
FF
|

n
=
0
dω.
(21)
In other words, at the quantum ground state, the contribution
of the quantum correlation
S
zF
to the readout spectrum
S
yy
has the same magnitude as that of the zero-point fluctuation,
while the sign of the correlation term depends on the sign of
the detuning of the readout beam.
This means that not only
has the experiment probed the zero-point fluctuation of the
mechanical oscillator, but it has also demonstrated nontrivial
correlations between the sensing noise and back-action noise
at the quantum level
.
C. Detailed theoretical analysis
In this section, we supply a detailed calculation of the
quantum dynamics and the output spectrum of the experiment.
The dynamics for a typical linear optomechanical device has
been studied extensively in the literature [
31
33
]; however, the
focus has been on the quantum state of the mechanical oscil-
lator in ground-state cooling experiments, instead of treating
the optomechanical device as a measurement device. Here we
will follow Ref. [
34
] and derive the corresponding input-output
relation—the analysis is the same as that of quantum noise in
a detuned signal-recycling laser interferometer, which can be
mapped into a detuned cavity [
15
,
35
,
36
]. We will focus only
on the interaction between the readout cavity mode and the
mechanical oscillator—the cooling mode and the thermal heat
bath are taken into account by the effective dynamics of the
oscillator, as mentioned earlier.
The Hamiltonian of our optomechanical system can be
written as [
31
33
]
ˆ
H
=
̄
r
ˆ
a
ˆ
a
+
ˆ
H
κ
r
+
̄
hG
0
ˆ
x
ˆ
a
ˆ
a
+
ˆ
p
2
2
m
+
1
2
2
m
ˆ
x
2
+
ˆ
H
κ
m
.
(22)
Here the first two terms describe the cavity mode including
its coupling to the external continuum, the third term is
the coupling between the cavity mode and the mechanical
oscillator,
G
0
=
ω
r
/L
c
is the coupling constant with
L
c
the
cavity length, and the rest of the terms describe the dynamics
of the effective oscillator (left and middle boxes in Fig.
2
), with
ˆ
H
κ
m
summarizing the dynamics of the cooling mode and the
thermal heat bath, as well as their coupling with the original
mechanical oscillator.
In the rotating frame at the laser frequency, the
linearized
equations of motion for the perturbed part—the variation
around the steady-state amplitude—read
m
(
̈
ˆ
x
+
κ
m
̇
ˆ
x
+
ω
2
m
ˆ
x
)
=
ˆ
F
BA
+
ˆ
F
q
,
(23)
̇
ˆ
a
+
(
κ
r
/
2
+
i
)
ˆ
a
=−
i
̄
G
0
ˆ
x
+
κ
r
ˆ
a
in
,
(24)
where the back-action force
ˆ
F
BA
is defined as
ˆ
F
BA
≡−
̄
h
̄
G
0
(
ˆ
a
+
ˆ
a
)
,
(25)
and we introduce
̄
G
0
=
̄
aG
0
, where
̄
a
is the steady-state
amplitude of the cavity mode and
ˆ
a
in
is the annihilation
operator of the input vacuum field. The cavity output
ˆ
a
out
is
related to the cavity mode by
ˆ
a
out
=−
ˆ
a
in
+
κ
r
ˆ
a,
(26)
with
κ
r
the decay rate (the bandwidth) of the readout mode.
In the steady state, these equations of motion can be
solved more easily in the frequency domain. Starting from
the mechanical displacement, we get
ˆ
x
(
ω
)
=
χ
(
ω
)[
ˆ
F
BA
(
ω
)
+
ˆ
F
q
(
ω
)]
.
(27)
Here we have ignored the modification to the mechanical
response function
χ
due to the readout mode—a term
proportional to
̄
G
2
0
, assuming that the pumping power is low.
For the cavity mode, we invert Eq.
(24)
and obtain
ˆ
a
(
ω
)
=
̄
G
0
ˆ
x
(
ω
)
+
i
κ
r
ˆ
a
in
(
ω
)
ω

+
r
/
2
,
(28)
which leads to
ˆ
F
BA
(
ω
)
=
2 ̄
h
̄
G
0
κ
r
/
2[(
κ
r
/
2
v
1
+

ˆ
v
2
]
(
ω

+
r
/
2)(
ω
+

+
r
/
2)
,
(29)
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QUANTUM BACK-ACTION IN MEASUREMENTS OF ZERO-
...
PHYSICAL REVIEW A
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, 033840 (2012)
with ˆ
v
1
(
ˆ
a
in
+
ˆ
a
in
)
/
2 and
v
2
(
ˆ
a
in
ˆ
a
in
)
/
(
2
i
) being
the amplitude quadrature and the phase quadrature of the
input field, which has fluctuations at the vacuum level. When
combining with Eq.
(26)
, we obtain the output amplitude
quadrature
ˆ
Y
1
(
ω
)
=
[
ˆ
a
out
(
ω
)
+
ˆ
a
out
(
ω
)]
/
2
=
(

2
κ
2
r
/
4
ω
2
)
ˆ
v
1
κ
r

ˆ
v
2
+
2
κ
r
̄
G
0

ˆ
x
(
ω

+
r
/
2)(
ω
+

+
r
/
2)
,
(30)
whose spectrum is measured experimentally. We put the above
formula into the same format as Eq.
(13)
by normalizing
ˆ
Y
1
with respect to the mechanical displacement
ˆ
x
, and introduce
ˆ
y
(
ω
) and the corresponding sensing noise
ˆ
z
(
ω
):
ˆ
y
(
ω
)
=
(

2
κ
2
r
/
4
ω
2
)
ˆ
v
1
κ
r

ˆ
v
2
2
κ
r
̄
G
0

+
ˆ
x
(
ω
)
ˆ
z
(
ω
)
+
χ
(
ω
)[
ˆ
F
BA
(
ω
)
+
ˆ
F
q
(
ω
)]
.
(31)
Taking the single-sided symmetrized spectral density of
ˆ
y
(see Appendix
A
), we obtain
S
yy
(
ω
)
=
S
zz
+
2Re[
χ
S
zF
]
+|
χ
|
2
[
S
BA
FF
+
S
q
FF
]
,
(32)
where
S
zz
(
ω
)
=
(

2
κ
2
r
/
4
ω
2
)
2
+
κ
2
r

2
2
κ
r
̄
G
2
0

2
,
(33)
S
zF
(
ω
)
=
̄
h
(
κ
r
/
2
)

,
(34)
S
BA
FF
(
ω
)
=
2 ̄
h
2
̄
G
2
0
κ
r
(
κ
2
r
/
4
+
ω
2
+

2
)
(

2
κ
2
r
/
4
ω
2
)
2
+
κ
2
r

2
.
(35)
Here we have used

0
|
ˆ
v
j
(
ω
v
k
(
ω

)
|
0

sym
=
πδ
jk
δ
(
ω
ω

)(
j,k
=
1
,
2)
.
(36)
Indeed, only
S
zF
depends on the sign of detuning and
contributes to the asymmetry. In the resolved-sideband case
κ
r
ω
m
and choosing detuning
|

|=
ω
m
,
S
zF
can be
approximated as the one shown in Eq.
(18)
. For a weak
readout beam, we can ignore
S
BA
FF
, which is proportional to
̄
G
2
0
, and the output spectra around
ω
m
for the positive- and
negative-detuning cases can be approximated as
S
yy
(
ω
)
|

ω
m
κ
r
2
̄
G
2
0
+
̄
m
(2

n
+
1
1)
2
m
[(
ω
ω
m
)
2
+
(
κ
m
/
2)
2
]
.
(37)
As we can see, the contribution to output spectra from the
quantum correlation has the same magnitude as the zero-point
fluctuation of the mechanical oscillator, with a sign depending
on the detuning. One can then obtain the dependence of the
asymmetry factor
η
on

n

,asshowninEq.
(9)
.
Interestingly, even if the quantum back-action term
S
BA
FF
is
much smaller than
S
q
FF
and has been ignored, given the weak
readout mode used in the experiment, the asymmetry induced
by quantum correlation is always visible as long as

n

is
small. In addition, any optical loss in the readout mode only
contributes a constant noise background—that is symmetric
with respect to detuning—to the overall spectrum; therefore,
the asymmetry is very robust against optical loss, and it can
be observed without a quantum-limited readout mode, which
is the case in the experiment.
D. Connection with the scattering picture
In the above, we have been emphasizing the viewpoint of
position measurement and interpreting the asymmetry as due
to the quantum correlation between the sensing noise and the
back-action noise. Here we would like to show the connection
between this viewpoint and the scattering picture in Ref. [
24
]
that focuses on the photon-phonon coupling, and, in addition,
show how the spectra of the amplitude quadrature measured
in the experiment are related to the emission spectra of the
(anti-)Stokes photons that would be obtained if we instead
take a photon-counting measurement.
To illustrate these, we introduce the annihilation operator
ˆ
b
for the phonon through the standard definition,
ˆ
x
̄
h/
(2
m
)(
ˆ
b
+
ˆ
b
)
,
(38)
and it satisfies the commutator relation [
ˆ
b,
ˆ
b
]
=
1. In the
rotating frame at the laser frequency, the Hamiltonian in
Eq.
(22)
after linearization is given by
ˆ
H
=
̄
h
ˆ
a
ˆ
a
+
ˆ
H
κ
r
+
̄
h
̄
g
0
(
ˆ
a
+
ˆ
a
)(
ˆ
b
+
ˆ
b
)
+
̄
m
ˆ
b
ˆ
b
+
ˆ
H
κ
m
,
(39)
where
̄
g
0
̄
G
0
̄
h/
(2
m
). The third term is the photon-
phonon coupling:
ˆ
a
ˆ
b
describes the anti-Stokes process where
the absorption of a phonon is accompanied by the emission
of a higher-frequency photon, and
ˆ
a
ˆ
b
describes the Stokes
process where the emission of a phonon is accompanied by the
emission of a lower-frequency photon. The photon emission
rate of these two processes can be estimated by using the
Fermi’s golden rule
. Specifically, taking into account the finite
bandwidth for the photon and phonon due to coupling to the
continuum, the emission rate of the anti-Stokes photon at
ω
lr
+
ω
reads
AS
(
ω
)
=
̄
g
2
0
dτe
iωτ
D
(
ω
)

ˆ
b
(
τ
)
ˆ
b
(0)

=
̄
g
2
0
κ
m

n

D
(
ω
)
(
ω
ω
m
)
2
+
(
κ
m
/
2)
2
,
(40)
and the emission rate of the Stokes photon at
ω
lr
ω
reads
S
(
ω
)
=
̄
g
2
0
dτe
iωτ
D
(
ω
)

ˆ
b
(
τ
)
ˆ
b
(0)

=
̄
g
2
0
κ
m
(

n
+
1)
D
(
ω
)
(
ω
ω
m
)
2
+
(
κ
m
/
2)
2
.
(41)
Here the density of state for the photons is determined by the
cavity decay rate and detuning:
D
(
ω
)
κ
r
/
2
(
ω

)
2
+
(
κ
r
/
2)
2
.
(42)
Were the cavity bandwidth much larger than the mechanical
frequency
ω
m
, the density of state
D
(
ω
) would become flat
for frequencies around
±
ω
m
, and we would effectively have
a scenario that is similar to the free-space Raman scattering
as in those spectroscopic measurements of crystals [
30
]. By
making a photon-counting-type measurement of the emitted
033840-5
FARID YA. KHALILI
et al.
PHYSICAL REVIEW A
86
, 033840 (2012)
(anti-)Stokes photons, one could observe an asymmetric
spectrum with two peaks (sidebands) around
ω
r
±
ω
m
of which
the profiles are given by the above emission rates. This is
also the case for those emission and absorption spectroscopic
measurements in the ions and atoms trapping experiments
[
25
28
].
The situation of our experiment is, however, different
from the usual free-space Raman-scattering spectroscopic
measurement by the following two aspects: (i)
We are
operating in the resolved-sideband regime
, where the cavity
bandwidth is much smaller than the mechanical frequency
and the photon density of state is highly asymmetric for
positive and negative sideband frequencies depending on the
detuning. This basically dictates that we cannot measure two
sidebands simultaneously, and we have to take two separate
spectra by tuning the laser frequency. In the positive-detuning
case

=
ω
m
, the anti-Stokes sideband is enhanced, while the
Stokes sideband is highly suppressed, as the photon density of
state is peak around
ω
=
ω
m
; while in the negative-detuning
case

=−
ω
m
, the situation for these two sidebands swaps.
(ii)
We are using a heterodyne detection scheme instead of
photon counting
, where the outgoing field is mixed with
a large coherent optical field (reference light) before the
photodetector, to measure the output amplitude quadrature,
and the signal is linear proportional to the position of the
oscillator, as we mentioned earlier. Interestingly, there is a
direct connection between the spectra of amplitude quadrature
measured in the experiment and the photon emission spectra
that are obtained if making photon-counting measurements.
To show this connection, we use the fact that
[
ˆ
Y
1
(
ω
)
,
ˆ
Y
1
(
ω

)]
=
0
,
(43)
which is a direct consequence of [
ˆ
y
(
t
)
,
ˆ
y
(
t

)]
=
0(
ˆ
y
is equal
to
ˆ
Y
1
normalized with respect to the mechanical displacement
[cf. Eq.
(31)
]), and we have

ˆ
Y
1
(
ω
)
ˆ
Y
1
(
ω

)

sym
=
ˆ
Y
1
(
ω

)
ˆ
Y
1
(
ω
)

=
1
2
[

ˆ
a
out
(
ω

)
ˆ
a
out
(
ω
)

+
ˆ
a
out
(
ω

)
ˆ
a
out
(
ω
)

]
.
(44)
Take the positive-detuning case

=
ω
m
, for instance,
ˆ
a
out
(
ω
) contains mostly vacuum and negligible sideband
signals due to suppression of the Stokes sideband around
ω
lr
ω
m
by the cavity, namely,

ˆ
a
out
(
ω

)
ˆ
a
out
(
ω
)
≈
2
πδ
(
ω
ω

)
.
The second term gives the emission spectrum for the
output photons shown in Eq.
(40)
; therefore, the single-sided
spectral density of the output amplitude quadrature reads
S
Y
1
Y
1
(
ω
)
=
1
+
2
AS
(
ω
)
.
(45)
By normalizing the spectrum with respect to the mechanical
displacement, we have
S
yy
(
ω
)
|

=
ω
m
=
κ
r
2
̄
G
2
0
[1
+
2
AS
(
ω
)]
.
(46)
Similarly, by following the same line of thought, we get
S
yy
(
ω
)
|

=−
ω
m
=
κ
r
2
̄
G
2
0
[1
+
2
S
(
ω
)]
.
(47)
The above two equations give identical results to Eq.
(37)
.
Therefore, the output spectra obtained in our heterodyne de-
tection differ from those in the photon-counting measurement
only by a constant noise floor, which originates from vacuum
fluctuation of the amplitude quadrature. After subtracting this
noise floor, we simply recover the emission spectra obtained
from taking the photon-counting measurement.
III. GENERAL LINEAR MEASUREMENTS OF THE
ZERO-POINT FLUCTUATION
Based on the analysis of the specific experiment of Ref. [
24
]
in the previous section, here we comment on the general
features of linear quantum measurements involving reading out
zero-point fluctuation of a mechanical oscillator. We start by
discussing the nature of the zero-point mechanical fluctuation
in Sec.
III A
, proceed to a discussion of the measurements of
it in Sec.
III B
, and finally end in Sec.
III C
, which discusses
its effect on sensitivity for measuring external forces and the
connection to the SQL.
A. The nature of zero-point mechanical fluctuation
First of all, let us take a closer look at the nature of
the zero-point fluctuation of a realistic harmonic oscillator,
which consists of a mechanical mode with eigenfrequency
ω
m
and finite decay rate
κ
m
. Suppose we initially decouple the
oscillator from its environmental heat bath and turn on the
coupling at
t
=
0. In the Heisenberg picture, the position and
momentum of the oscillator at
t>
0 will be
ˆ
x
q
(
t
)
=
ˆ
x
free
(
t
)
+
t
0
χ
(
t
t

)
ˆ
F
q
(
t

)
dt

,
(48a)
ˆ
p
q
(
t
)
=
ˆ
p
free
(
t
)
+
m
t
0
̇
χ
(
t
t

)
ˆ
F
q
(
t

)
dt

,
(48b)
where
ˆ
x
free
(
t
)
=
e
κ
m
t/
2
[
ˆ
x
(0) cos
ω
m
t
+
ˆ
p
(0)
m
sin
ω
m
t
]
,
(49a)
ˆ
p
free
(
t
)
m
=
e
κ
m
t/
2
[
ˆ
x
(0) sin
ω
m
t
+
ˆ
p
(0)
m
cos
ω
m
t
]
m
2
ˆ
x
free
(
t
)
,
(49b)
are contributions from the free evolution of the initial
Schr
̈
odinger operators (i.e., undisturbed by the environment),
which decay over time and get replaced by contributions from
the environmental heat bath [integrals on the right-hand side of
Eqs.
(48a)
and
(48b)
]. Note that for any oscillator with nonzero
decay rate, it is essential to have bath operators entering over
time, otherwise the commutation relation between position and
momentum,
[
ˆ
x
q
(
t
)
,
ˆ
p
q
(
t
)]
=
i
̄
h,
(50)
will not hold at
t>
0 because of
[
ˆ
x
free
(
t
)
,
ˆ
p
free
(
t
)]
=
i
̄
he
κ
m
t
.
(51)
This dictates that the heat bath must be such that the additional
commutator from terms containing
ˆ
F
q
exactly compensates for
the decay in Eq.
(51)
, which leads to the quantum fluctuation-
dissipation theorem (see, e.g., Ref. [
37
]).
033840-6
QUANTUM BACK-ACTION IN MEASUREMENTS OF ZERO-
...
PHYSICAL REVIEW A
86
, 033840 (2012)
It is interesting to note that this “replenishing” of com-
mutators has a classical counterpart, since commutators are,
after all, proportional to the classical Poisson bracket. More
specifically, for a classical oscillator with decay, we can write
a similar relation for Poisson brackets among the position and
momentum of the oscillator, plus environmental degrees of
freedom. The replenishing of the position-momentum Poisson
bracket by environmental ones, in classical mechanics, can
also be viewed as a consequence of the conservation of phase-
space volume, following the Liouville theorem. A decaying
oscillator’s phase-space volume will shrink and violate the
Liouville theorem—unless additional phase-space volume
from the environmental degrees of freedom is introduced.
Nevertheless, the definitive quantum feature in our situation
is a fundamental scale in the volume of phase space, which is
equal to ̄
h
. Here we note that if
κ
m
ω
m
, when reaching the
steady state with
ˆ
x
free
and
ˆ
q
free
decayed away, then we have
x
q
p
q
m
2
π
S
q
xx
(
ω
)
,
(52)
where
S
q
xx
≡|
χ
|
2
S
q
FF
. Although
S
q
xx
depends on the spe-
cific scenario, they are all constrained by a Heisenberg-like
relation of
S
q
xx
(
ω
)

2 ̄
h
Im
χ
(
ω
)
,
(53)
which is a straightforward consequence of the commutation
relation in Eq.
(11)
. The equality is achieved at the ground
state.
2
This enforces the same Heisenberg uncertainty relation,
x
q
p
q

̄
h/
2
,
(54)
as an ideal harmonic oscillator whose quantum fluctuations
arise “on their own,” instead of having to be driven by the sur-
rounding environment. Therefore, in the steady state, the zero-
point fluctuation of the mechanical oscillator can be viewed
as being imposed by the environment due to the linearity of
the dynamics.
B. Measuring the zero-point fluctuation
Having clarified the nature of quantum zero-point fluctua-
tions of a mechanical oscillator in the steady state, let us argue
that the effects seen in Ref. [
24
] are actually generic when
one tries to probe such fluctuations, namely: the correlation
between sensing and back-action noise can be at the level of
the zero-point fluctuation itself.
Let us start our discussion here from Eq.
(5)
, namely,
[
ˆ
y
(
t
)
,
ˆ
y
(
t

)]
=
0
,
(55)
and the fact that
ˆ
y
consists of sensing noise, back-action
noise, and finally the zero-point fluctuation of the mechanical
oscillator [cf. Eq.
(8)
]:
ˆ
y
(
t
)
=
ˆ
z
(
t
)
α
+
α
t
−∞
χ
(
t
τ
)
ˆ
F
BA
(
τ
)
+
ˆ
x
q
(
t
)
.
(56)
Here we have added a factor
α
, which labels the scaling of each
term as the measurement strength which is proportional to the
2
A generalization of this to thermal states will be the fluctuation-
dissipation theorem [
37
].
square root of the readout beam power. Let us assume that the
dynamical response
χ
of the oscillator is not modified due to
couplings to the measurement field, and Eq.
(55)
continues to
hold for the same set of
ˆ
z
and
ˆ
F
BA
, for a large set of
α
and
χ
:
basically, the measuring device works for different mechanical
oscillators with different measuring strength.
Since Eq.
(55)
remains valid for all values of
α
, we extract
terms with different powers of scaling, and obtain
[
ˆ
z
(
t
)
,
ˆ
z
(
t

)]
=
[
ˆ
F
BA
(
t
)
,
ˆ
F
BA
(
t

)]
=
0
,
(57)
and
t

−∞
χ
(
t

τ
)[
ˆ
z
(
t
)
,
ˆ
F
BA
(
τ
)]
t
−∞
χ
(
t
τ
)[
ˆ
z
(
t

)
,
ˆ
F
BA
(
τ
)]
+
[
ˆ
x
q
(
t
)
,
ˆ
x
q
(
t

)]
=
0
t,t

.
(58)
This becomes
+∞
0
χ
(
τ
)
[
C
zF
(
t
τ
)
C
zF
(
t
τ
)
]
=−
i
̄
(
t
) (59)
for all values of
t
, where we have defined
C
zF
(
t

t
)
[
ˆ
z
(
t
)
,
ˆ
F
BA
(
t

)]
.
(60)
Here the dependence is only through
t

t
because the system
is assumed to be time invariant. We also note that since
ˆ
z
is an outgoing field,
C
zF
(
t

t
) must vanish when
t

t>
0, otherwise any generalized force applied on the outgoing
field
ˆ
z
(
t
), detached from the mechanical oscillator, can still
dynamically influence the mechanical motion at later times
(future) through
ˆ
F
BA
(
t

), which violates the causality [
11
,
35
].
As proven in Appendix
B
, in order for Eq.
(59)
to be satisfied
for all possible response functions of the oscillator, we must
have
C
zF
(
t
)
=−
i
̄
(
t
)
,
(61)
where
δ
(
t
)istheDirac
δ
function with support only for
t<
0.
In other words,
[
ˆ
z
(
t
)
,
ˆ
F
BA
(
t

)]
=−
i
̄
(
t

t
)
.
(62)
Equation
(56)
, plus the commutation relations in Eqs.
(57)
and
(62)
, then provide a general description of linear measuring
devices, which do not modify the dynamics of the mechanical
oscillator—simply from the requirement that the outgoing field
operators at different times must commute [cf. Eq.
(55)
].
In particular, the nonvanishing commutator
[
ˆ
x
q
(
t
)
,
ˆ
x
q
(
t

)
]
,
which underlies the existence of the zero-point fluctuation,
is canceled in a simple way by the nonvanishing commu-
tator between the sensing noise and the back-action noise
[cf. Eq.
(62)
].
Now turn to the noise content of the output
ˆ
y
(
t
), i.e., the
spectrum
S
yy
=
S
zz
α
2
+
2Re[
χ
S
zF
]
+
α
2
S
BA
FF
+
S
q
xx
.
(63)
Let us consider experiments with relatively low measurement
strength, so that the first term
S
zz
2
from the sensing noise
dominates the output noise. The next-order terms contain (i)
correlation between the sensing noise and the back-action
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