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, CA 91125, USA
3
Thomas J. Watson, Sr., Laboratory of Applied Physics,
California Institute of Technology, Pasadena, California 91125, USA
Measurement-induced back action, a direct consequence of the Heisenberg Uncertainty Principle, is the defin-
ing 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 results of this experiment not
only characterize the zero-point fluctuation of a near-ground-state nanomechanical oscillator, but also demon-
strate 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.
PACS numbers:
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 Prin-
ciple
[
ˆ
x
,
ˆ
p
] =
i
̄
h
,
(1)
which leads to:
∆
x
·
∆
p
≥
̄
h
/
2
.
(2)
Here ˆ
x
and ˆ
p
are the position and momentum operators, while
∆
x
and
∆
p
are standard deviations of position and momen-
tum for an arbitrary quantum state. Eq. (2) means we cannot
specify the position and momentum of a harmonic oscillator
simultaneously, as a point in classical phase space — the os-
cillator 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
ω
m
,
(3)
which leads to:
∆
x
q
(
t
)
·
∆
x
q
(
t
′
)
≥
̄
h
|
sin
ω
m
(
t
′
−
t
)
|
2
m
ω
m
.
(4)
with ˆ
x
q
(
t
)
being the Heisenberg operator of the oscillator po-
sition, quantum-mechanically evolving under the free Hamil-
tonian. Here
∆
x
q
(
t
)
is the standard deviation of ˆ
x
q
(
t
)
for an
arbitrary quantum state. Eq. (4) means the position of a freely
evolving quantum harmonic oscillator cannot continuously as-
sume precise values, but instead, must fluctuate. This fluctua-
tion carries the zero-point mechanical energy of
̄
h
ω
m
/
2.
As a key feature of quantum mechanics, zero-point fluctua-
tion 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 observa-
tion 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. Eqs. (3) and
(4), which argue for the inevitability of the zero-point fluc-
tuation, 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 experi-
mental 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
ε
(
t
)+
ˆ
x
q
(
t
)
(6)
with non-vanishing additional noise (error)
ˆ
ε
(
t
)
, which con-
sists of degrees of freedom of the measurement device and
compensates the non-vanishing commutator of ˆ
x
q
‡
. In addi-
tion, during the measurement process, 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
‡
We note that Ozawa has developed a different formalism to quantify the is-
sues that arise when attempts are made to measure non-commuting observ-
ables like ˆ
x
q
(
t
)
[9, 10]. However, we have chosen to adopt the Braginsky-
Khalili approach [11], because it is immediately applicable when the non-
commuting observable is acting as a probe for an external classical force.
arXiv:1206.0793v1 [quant-ph] 4 Jun 2012
2
external observable
ˆ
F
, which introduces a term proportional
to ˆ
x
ˆ
F
into the Hamiltonian, will have to cause non-zero dis-
turbance. For this reason, we can expand the measurement er-
ror
ˆ
ε
into two parts: ˆ
z
— the sensing noise that is independent
from mechanical motion and ˆ
x
BA
— 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
ε
(
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 Bra-
ginsky in the late 1960s aiming at describing 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 measure-
ment theory is the trade-off between sensing noise and back-
action noise, which gives rise to the so-called Standard Quan-
tum Limit (SQL). For optomechanical devices, sensing noise
takes the form of quantum shot noise due to discreteness of
photons, while the quantum back-action is enforced by quan-
tum fluctuations in the radiation pressure acting on the me-
chanical oscillators [13], which is therefore also called quan-
tum 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 metrol-
ogy, e.g., gravitational-wave detections [14–22]. At the mo-
ment, it is still experimentally challenging to directly observe
quantum radiation-pressure noise in optomechanical devices
due to high levels of environmental thermal fluctuations, 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 im-
mune to thermal fluctuations.
In this article, 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 oscilla-
tion. 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. Using 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 non-trivial correlations between
sensing noise and back-action noise — an much sought-after
effect in the gravitational-wave-detection community.
Its con-
tribution to the output spectrum is equal to the zero-point fluc-
tuation for one detuning of the readout beam, and exactly op-
posite for the other detuning.
The outline of this article goes 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 mechanical zero-point fluctuation, show that in
attempts to measure the zero-point fluctuation, the contribu-
tions from sensing–back-action noise correlations can generi-
cally be comparable to the zero-point fluctuation itself. In ad-
dition, 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
In this section, 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
will 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 cou-
pled to a mechanical vibrational mode in a patterned sili-
con nanobeam. One spatial mode — the cooling mode —
is pumped with a relatively high power at a “red” detun-
ing (lower than resonance), and is used to cool the mechan-
ical mode via radiation pressure damping [2]; the other cav-
ity 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
noise floor
noise floor
FIG. 1: Figure illustrating the observed spectra of the readout laser
in the positive-detuning case (left) and the negative-detuning case
(right).
3
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 fre-
quency
, 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. In Ref. [24], we introduced the following
figure of merit to quantify the asymmetry:
η
≡
I
−
I
+
−
1
=
1
〈
n
〉
.
(9)
We interpreted this asymmetry as arising from the quan-
tized motion of the mechanical oscillator. The asymmetry is
thus assigned to the difference between the phonon absorption
rate, proportional
〈
n
〉
, and the emission rate, proportional to
〈
n
〉
+
1. This is completely analogous to that used for calibra-
tion of motional thermometry of ions/atoms trapped in electri-
cal/optical traps [25–28]. Additionally, these scattering pro-
cesses have an underlying physics similar to bulk nonlinear
Raman scattering processes used in spectroscopic analysis of
crystals [29, 30], where an ensemble of vibrational degrees of
freedom internal to the molecular structure of the system in-
teracts 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 het-
erodyne 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, spec-
tra of the amplitude quadrature are equal to 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-
Stoke 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], em-
phasizing on 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 cooling
mode, the mechanical oscillator, and the environmental ther-
mal bath the oscillator couples to (the left and middle boxes in
Fig. 2) together is the first part, which can be viewed as pro-
viding an
effective mechanical oscillator
nearly at the ground
state, but with a quality factor significantly lower than the in-
trinsic 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
mechanical
oscillator
thermal heat
bath
&
cooling mode
readout
mode
FIG. 2: Figure illustrating 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.
effective oscillator (the first part of the system) through dis-
placement 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.
1. The Mechanical Oscillator Near Ground State
Let us focus on the first part of the system (left two boxes
of Fig. 2), the effective mechanical oscillator (Because 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 steady
state, the “free” mechanical displacement is determined by its
coupling to this bath (“free” means in absence of the readout
mode):
ˆ
x
q
(
t
) =
∫
t
−
∞
χ
(
t
−
t
′
)
ˆ
F
q
(
t
′
)
d
t
′
.
(10)
Here
χ
is the response function of the mechanical oscillator:
χ
(
t
−
t
′
) =
−
[
ˆ
x
(
t
)
,
ˆ
x
(
t
′
)]
i
̄
h
=
e
−
κ
m
|
t
−
t
′
|
/
2
sin
ω
m
(
t
−
t
′
)
m
ω
m
.
(11)
Note that we have an additional decay factor compared with
Eq. (3) which describes an
idealized
free oscillator. The de-
cay rate
κ
m
here is determined jointly by the intrinsic decay
rate of the mechanical mode, and the optomechanical interac-
tion between the mechanical mode and the cooling mode. The
force
ˆ
F
q
lumps together the fluctuating force acting on the me-
chanical mode by the environmental heat bath and the cooling
mode. If the oscillator approaches the ground state only af-
ter 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 mechani-
cal oscillator with non-zero decay rate, its zero-point fluctua-
tion in the steady state can be viewed as driven by the quantum
heat bath surrounding it. We will returning 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 po-
sition meter that measures the mechanical displacement. We
4
can rewrite the disturbance ˆ
x
BA
in Eq. (8) in terms of the back-
action force
ˆ
F
BA
arising from radiation-pressure fluctuation of
the readout mode, namely,
ˆ
x
BA
(
t
) =
∫
t
−
∞
χ
(
t
−
t
′
)
ˆ
F
BA
(
t
′
)
d
t
′
.
(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
+
i
κ
m
ω
)
(14)
is the Fourier transform of
Θ
(
t
)
χ
(
t
)
, with
Θ
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
F F
+
|
χ
|
2
S
q
F F
.
(15)
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 self spectrum;
S
zz
and
S
BA
F F
are the sensing-noise and back-action force noise spec-
trum, respectively;
S
zF
is the cross correlation between ˆ
z
and
ˆ
F
BA
; 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
F F
= (
4
〈
n
〉
+
2
)
̄
hm
κ
m
ω
m
,
(16)
and
〈
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
6
=
S
yy
(
ω
)
|
∆
=
ω
m
.
(17)
As we will show in the Sec. II C that follows, when we flip the
sign of the detuning
∆
of the readout beam, the only term in
S
yy
that changes is
S
zF
— the correlation between the sensing
noise and the back-action noise. According to Eq. (34), we
have
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
)]
d
ω
∫
[
|
χ
|
2
S
q
F F
+
2Re
(
χ
∗
S
+
zF
)
]
d
ω
=
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
con-
tribute to the above integral,
S
±
zF
≈∓
i
̄
h
.
(20)
The asymmetry, or effect of quantum correlation
S
zF
, is
most prominent when the thermal occupation number ap-
proaches zero. Indeed, if we focus on the quantum fluctuation
by taking the limit of
〈
n
〉→
0, we obtain
∫
2Re
(
χ
∗
S
±
zF
)
d
ω
=
∓
∫
|
χ
|
2
S
q
F F
|
〈
n
〉
=
0
d
ω
.
(21)
In other words, at the quantum ground state, 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 de-
tuning of the readout beam.
This means not only has the ex-
periment probed the zero-point fluctuation of the mechanical
oscillator, it has also demonstrated non-trivial correlations
between sensing noise and back-action noise at the quantum
level
.
C. Detailed theoretical analysis
In this section, we supply a detailed calculation of the quan-
tum 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, they
have been focusing on quantum state of the mechanical os-
cillator in ground-state cooling experiments, instead of treat-
ing 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 the one of quan-
tum 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 is 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
=
̄
h
ω
r
ˆ
a
†
ˆ
a
+
ˆ
H
κ
r
+
̄
h G
0
ˆ
x
ˆ
a
†
ˆ
a
+
ˆ
p
2
2
m
+
1
2
m
ω
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 cou-
pling between the cavity mode and the mechanical oscillator;
G
0
=
ω
r
/
L
c
is the coupling constant with
L
c
the cavity length;
the rest of the terms describes the dynamics of the effective
oscillator (left and middle boxes in Fig. 2), with
ˆ
H
κ
m
summa-
rizing 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 — variation around
5
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
with ̄
a
being the steady-state am-
plitude 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 me-
chanical displacement, we get
ˆ
x
(
ω
) =
χ
(
ω
)[
ˆ
F
BA
(
ω
)+
ˆ
F
q
(
ω
)]
.
(27)
Here we have ignored modification to the mechanical re-
sponse function
χ
due to the readout mode—a term propor-
tional 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
−
∆
+
i
κ
r
/
2
,
(28)
which leads to
ˆ
F
BA
(
ω
) =
2
̄
h
̄
G
0
√
κ
r
/
2
[(
κ
r
/
2
−
i
ω
)
ˆ
v
1
+
∆
ˆ
v
2
]
(
ω
−
∆
+
i
κ
r
/
2
)(
ω
+
∆
+
i
κ
r
/
2
)
,
(29)
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 com-
bining with Eq. (26), we obtain the output amplitude quadra-
ture
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
−
∆
+
i
κ
r
/
2
)(
ω
+
∆
+
i
κ
r
/
2
)
,
(30)
whose spectrum is measured experimentally.
We put the
above formula into the same format as Eq.(13) by normal-
izing
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 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
F F
+
S
q
F F
]
,
(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
−
i
ω
)
∆
,
(34)
S
BA
F F
(
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
)
ˆ
v
†
k
(
ω
′
)
|
0
〉
sym
=
π δ
jk
δ
(
ω
−
ω
′
) (
j
,
k
=
1
,
2
)
.
(36)
Indeed, only
S
zF
depends on the sign of detuning and con-
tributes to the asymmetry.
In the resolved-sideband case
κ
r
ω
m
and choosing detuning
|
∆
|
=
ω
m
,
S
zF
can be ap-
proximated as the one shown in Eq. (18). For a weak readout
beam, we can ignore
S
BA
F F
which is proportional to
̄
G
2
0
, the out-
put 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
+
̄
h
κ
m
(
2
〈
n
〉
+
1
∓
1
)
2
m
ω
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
〉
shown in Eq. (9).
Interestingly, even if the quantum back-action term
S
BA
F F
is
much smaller than
S
q
F F
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 spectra of the amplitude quadrature measured in the
experiment are related to emission spectra of the (anti-)Stokes
photons that would have been 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
ω
m
)(
ˆ
b
+
ˆ
b
†
)
.
(38)
6
and it satisfies the commutator relation:
[
ˆ
b
,
ˆ
b
†
] =
1. In the ro-
tating 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
†
)+
̄
h
ω
m
ˆ
b
†
ˆ
b
+
ˆ
H
κ
m
,
(39)
where ̄
g
0
≡
̄
G
0
√
̄
h
/
(
2
m
ω
m
)
. The third term is the photon-
phonon coupling: ˆ
a
†
ˆ
b
describes the anti-Stokes process —
the absorption of a phonon is accompanied by emission of a
higher-frequency photon; ˆ
a
†
ˆ
b
†
describes the Stokes process
— the emission of a phonon is accompanied by 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:
Γ
AS
(
ω
) =
̄
g
2
0
∫
d
τ
e
i
ωτ
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)
the emission rate of the Stokes photon at
ω
lr
−
ω
reads:
Γ
S
(
ω
) =
̄
g
2
0
∫
d
τ
e
−
i
ωτ
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
(anti-)Stokes photons, one could observe an asymmetric spec-
trum 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 mea-
surements in the ions/atoms trapping experiments [25–28].
The situation of our experiment is however different from
the usual free-space Raman scattering spectroscopic measure-
ment 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 basi-
cally dictates that we cannot measure two sidebands simulta-
neously, 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 side-
band 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 heterodyne detection scheme instead of photon
counting
, where the outgoing field is mixed with a large co-
herent 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 con-
nection, 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
signal 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
π δ
(
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
Γ
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
Γ
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
Γ
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 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 fea-
tures of linear quantum measurements involving reading out
zero-point fluctuation of a mechanical oscillator. We start
from discussing nature of the zero-point mechanical fluctu-
ation in Sec. III A, proceed to discussion of 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.
7
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 con-
sists 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 Heinserberg 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
′
)
d
t
′
,
(48a)
ˆ
p
q
(
t
) =
ˆ
p
free
(
t
)+
m
∫
t
0
̇
χ
(
t
−
t
′
)
ˆ
F
q
(
t
′
)
d
t
′
,
(48b)
where
ˆ
x
free
(
t
) =
e
−
κ
m
t
/
2
[
ˆ
x
(
0
)
cos
ω
m
t
+
ˆ
p
(
0
)
m
ω
m
sin
ω
m
t
]
,
(49a)
ˆ
p
free
(
t
)
m
ω
m
=
e
−
κ
m
t
/
2
[
−
ˆ
x
(
0
)
sin
ω
m
t
+
ˆ
p
(
0
)
m
ω
m
cos
ω
m
t
]
−
m
κ
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 non-
zero decay rate, it is essential to have bath operators entering
over time, otherwise the commutation relation between posi-
tion 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
̄
h e
−
κ
m
t
.
(51)
This dictates that the heat bath must be such that the additional
commutator from terms containing
ˆ
F
q
exactly compensate for
the decay in Eq. (51), which leads to the quantum fluctuation-
dissipation theorem (see e.g., Ref. [37]).
It is interesting to note that this “replenishing” of commu-
tators has a classical counterpart, since commutators are after
all proportional to the classical Poisson Bracket. More specif-
ically, for a classical oscillator with decay, we can write a sim-
ilar relation for Poisson Brackets among the position and mo-
mentum of the oscillator, plus environmental degrees of free-
dom. 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, we have
∆
x
q
·
∆
p
q
≈
m
ω
m
∫
d
ω
2
π
S
q
xx
(
ω
)
,
(52)
where
S
q
xx
≡|
χ
|
2
S
q
F F
. Although
S
q
xx
depends on the specific
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
§
. This enforces the same Heisenberg Uncertainty rela-
tion:
∆
x
q
·
∆
p
q
≥
̄
h
/
2
,
(54)
as an ideal harmonic oscillator whose quantum fluctuations
arise “on its 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 linear-
ity 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 be-
tween 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 oscil-
lator [cf. Eq. (8)]:
ˆ
y
(
t
) =
ˆ
z
(
t
)
α
+
α
∫
t
−
∞
χ
(
t
−
τ
)
ˆ
F
BA
(
τ
)
d
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 square root of the readout beam power. Let us assume
that 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 mea-
suring 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)
§
A generalization of this to thermal states will be the fluctuation-dissipation
theorem [37].
8
and
∫
t
′
−
∞
χ
(
t
′
−
τ
)
[
ˆ
z
(
t
)
,
ˆ
F
BA
(
τ
)
]
d
τ
−
∫
t
−
∞
χ
(
t
−
τ
)
[
ˆ
z
(
t
′
)
,
ˆ
F
BA
(
τ
)
]
d
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
)]
d
τ
=
−
i
̄
h
χ
(
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
assumed to be time-invariant. We also note that since ˆ
z
is
an out-going field,
C
zF
(
t
′
−
t
)
must vanish when
t
′
−
t
>
0,
otherwise any generalized force applied on the out-going 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 proved in the App. B, in order for Eq. (59) to be satisfied
for all possible response functions of the oscillator, we must
have
C
zF
(
t
) =
−
i
̄
h
δ
−
(
t
)
,
(61)
where
δ
−
(
t
)
is the Dirac delta function with support only for
t
<
0. In other words,
[
ˆ
z
(
t
)
,
ˆ
F
BA
(
t
′
)] =
−
i
̄
h
δ
−
(
t
′
−
t
)
.
(62)
Eq. (56), plus the commutation relations in Eqs. (57) and
(62), then provides a general description of linear measur-
ing devices which do not modify the dynamics of the me-
chanical oscillator — simply from the requirement that the
out-going field operators at different times must commute
[cf.
Eq. (55)].
In particular, non-vanishing 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 non-vanishing
commutator 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
F F
+
S
q
xx
.
(63)
Let us consider experiments with relatively low measurement
strength, so that the first term
S
zz
/
α
2
from the sensing noise
dominates the output noise. The next-order terms contain:
(i) correlation between the sensing noise and the back-action
noise —
S
zF
; and (ii) the mechanical fluctuation —
S
q
xx
. If we
assume nearly ground state for the mechanical oscillator
S
q
xx
(
ω
)
≈
2
̄
h
Im
χ
(
ω
)
,
(64)
which, for
κ
m
ω
m
, gives
∫
d
ω
2
π
S
q
xx
(
ω
)
≈
̄
h
2
m
ω
m
.
(65)
If
S
zF
(
ω
)
does not change noticeably within the mechanical
bandwidth, then
∫
d
ω
2
π
2Re
[
χ
∗
(
ω
)
S
zF
(
ω
)]
≈−
1
2
m
ω
m
Im
S
zF
(
ω
m
)
.
(66)
Because of Eq. (62), the typical magnitude for
S
zF
is natu-
rally
¶
|
S
zF
|∼
̄
h
.
(67)
Therefore, contributions to the output noise from quantum
correlation
S
zF
and mechanical fluctuation
S
q
xx
can generically
become comparable to each other when the mechanical os-
cillator is approaching the quantum ground state. The result
presented in Ref. [24] therefore illustrates two typical cases of
this generic behavior [cf. Eq. (20)].
C. Measuring external classical forces in presence of
zero-point fluctuation
Finally, let us discuss the role of zero-point fluctuation in
force measurement, when the mechanical oscillator is used
as a probe of external classical forces not far away from the
mechanical resonant frequency. The force sensitivity of such a
linear measurement device, in terms of spectral density
S
F
, is
obtained by normalizing the displacement sensitivity
S
yy
with
respect to the mechanical response function
χ
:
S
F
≡
S
yy
/
|
χ
|
2
.
Specifically, from Eq. (15), we have
S
F
(
ω
) =
S
zz
(
ω
)
|
χ
(
ω
)
|
2
+
2Re
[
S
zF
(
ω
)
χ
(
ω
)
]
+
S
BA
F F
(
ω
)+
S
q
F F
(
ω
)
.
(68)
Because of the commutation relations in Eqs. (57) and (62),
a Heisenberg Uncertainty Relation exists among the spectral
densities of ˆ
z
and
ˆ
F
BA
, and that is
S
zz
(
ω
)
S
BA
F F
(
ω
)
−
S
zF
(
ω
)
S
F z
(
ω
)
≥
̄
h
2
.
(69)
When the the sensing noise ˆ
z
and the back-action noise
ˆ
F
BA
are
not correlated
—
S
zF
=
S
F z
=
0, we have
S
zz
(
ω
)
S
BA
F F
(
ω
)
≥
̄
h
2
.
(70)
The above inequality represents a trade-off between sensing
noise ˆ
z
and back-action noise
ˆ
F
BA
. Correspondingly, the force
sensitivity will have a lower bound :
S
F
(
ω
)
|
S
zF
=
0
=
S
zz
(
ω
)
|
χ
(
ω
)
|
2
+
S
BA
F F
(
ω
)+
S
q
F F
(
ω
)
≥
2
̄
h
|
χ
(
ω
)
|
+(
4
〈
n
〉
+
2
)
̄
h m
κ
m
ω
m
.
(71)
¶
In general, the commutator does not impose any bound on the cross corre-
lation. Here, in a strict sense, is an order-of-magnitude estimate.