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Laser noise in cavity-optomechanical cooling and thermometry
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Laser noise in cavity-optomechanical
cooling and thermometry
Amir H Safavi-Naeini
1
,
5
, Jasper Chan
1
, Jeff T Hill
1
,
Simon Gr
̈
oblacher
1
,
2
, Haixing Miao
3
, Yanbei Chen
3
,
Markus Aspelmeyer
4
and Oskar Painter
1
,
5
1
Kavli Nanoscience Institute and Thomas J Watson, Sr, Laboratory of Applied
Physics, California Institute of Technology, Pasadena, CA 91125, USA
2
Institute for Quantum Information and Matter, California Institute of
Technology, 1200 E California Boulevard, Pasadena, CA 91125, USA
3
Theoretical Astrophysics 350-17, California Institute of Technology,
Pasadena, CA 91125, USA
4
Vienna Center for Quantum Science and Technology (VCQ), Faculty of
Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
E-mail:
safavi@caltech.edu
and
opainter@caltech.edu
New Journal of Physics
15
(2013) 035007 (35pp)
Received 10 October 2012
Published 6 March 2013
Online at
http://www.njp.org/
doi:10.1088/1367-2630/15/3/035007
Abstract.
We review and study the roles of quantum and classical fluctuations
in recent cavity-optomechanical experiments which have now reached the
quantum regime (mechanical phonon occupancy
.
1) using resolved sideband
laser cooling. In particular, both the laser noise heating of the mechanical
resonator and the form of the optically transduced mechanical spectra, modified
by quantum and classical laser noise squashing, are derived under various
measurement conditions. Using this theory, we analyze recent ground-state
laser cooling and motional sideband asymmetry experiments with nanoscale
optomechanical crystal resonators.
5
Authors to whom any correspondence should be addressed.
C
ontent from this work may be used under the terms of the
Creative Commons Attribution-NonCommercial-
ShareAlike 3.0 licence
. Any further distribution of this work must maintain attribution to the author(s) and the title
of the work, journal citation and DOI.
New Journal of Physics
15
(2013) 035007
1367-2630/13/035007+35
$
33.00
© IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
Contents
1. Introduction
2
2. Theory
3
2.1. Quantum-limited laser cooling and damping
. . . . . . . . . . . . . . . . . . .
5
2.2. Thermometry with the cooling beam
. . . . . . . . . . . . . . . . . . . . . . .
6
2.3. Motional sideband asymmetry thermometry
. . . . . . . . . . . . . . . . . . .
8
2.4. Laser phase noise
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
3. Experiment
15
3.1. Experimental setup
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.2. Laser cooling and cooling beam mode thermometry
. . . . . . . . . . . . . . .
18
3.3. Sideband asymmetry mode thermometry
. . . . . . . . . . . . . . . . . . . . .
22
4. Conclusions
25
Acknowledgments
25
Appendix A. Definitions
25
Appendix B. Mechanical resonator spectral density
26
Appendix C. Quantum noise squashing
27
Appendix D. Scattering matrix elements
28
Appendix E. Laser phase noise measurement
28
References
32
1. Introduction
Mechanical resonators provide an interesting and useful system for the study of quantum
theory in the mesoscopic and macroscopic scales [
1
6
]. Technological progress in nano- and
microfabrication have made coupling of motion to a wide variety of systems possible, and
experiments demonstrating mechanical resonators coupled to optical cavities [
7
16
], microwave
resonators [
17
20
], superconducting qubits [
21
23
], cold atoms [
24
], and defect centers in
diamond [
25
] have been forthcoming. These advances have led to the cooling of mechanical
systems to their quantum ground states [
19
,
22
,
26
], and the observation of non-classical
behavior [
22
,
27
,
28
]. Of all the systems to which mechanical systems have been shown to
couple, it has long been realized [
1
] that light has many distinct advantages. Most importantly
light sources can be made quantum limited (lasers) and lack thermal noise even at room
temperature. This allows, for example, the cooling of optomechanical systems close to their
ground states using radiation pressure forces, and recently experiments starting with modest
cryogenic pre-cooling [
14
,
26
,
27
,
29
] have been successful at achieving this task.
Considering the importance of cooling and thermometry in cavity-optomechanical
experiments, it is important to understand the noise processes which can lead to both heating and
systematic errors in thermometry. In this paper we focus on analyzing these effects theoretically,
and compare to recently performed experiments with nanoscale optomechanical crystal (OMC)
devices. In section
2
a theoretical treatment of optomechanical cooling and thermometry
particularly suited for understanding the propagation of noise is given. The regime of operation
analyzed is that relevant to our recent experiments [
26
,
27
], i.e. the driven weak-coupling,
sideband-resolved regime where the cavity decay rate is larger than all other rates in the system
New Journal of Physics
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3
except for the mechanical frequency (a more general treatment of phase noise, including the case
of sideband-unresolved systems was presented recently in [
30
]). In this regime, the standard
input–output formalism [
31
33
] for analyzing the linearized system is applied. Two different
methods of thermometry used in recent experiments [
26
,
27
] are treated in sections
2.2
and
2.3
,
respectively. The former involves directly measuring the light scattered by mechanical motion
and calibrating its intensity given a set of system parameters, while the latter requires comparing
the emission and absorption rates of phonons from mechanical subsystems, and observing a non-
classical asymmetry analogous to that seen in much earlier experiments with trapped atoms
and ions [
34
36
]. Both methods are susceptible to laser phase noise, which is discussed in
section
2.4
, through both noise induced heating [
37
40
], and systematic errors in thermometry
caused by noise
squashing
[
18
,
41
] and
anti-squashing
. Finally, in section
3
we review recent
laser cooling and sideband asymmetry measurements of OMC cavities near the quantum ground
state of their mechanical motion, and compare these results with the measured phase noise (see
appendix
E
) of the external cavity semiconductor diode lasers used in these experiments.
2. Theory
The optomechanical interaction between a mechanical system and an optical field occurs
through radiation pressure, a force proportional to the optical field intensity. This can be
modeled by a Hamiltonian
H
=
̄
h
ω
o
ˆ
a
ˆ
a
+
̄
h
ω
m0
ˆ
b
ˆ
b
+
̄
hg
0
ˆ
a
ˆ
a
(
ˆ
b
+
ˆ
b
),
(1)
with
ˆ
a
and
ˆ
b
the annihilation operators for photons and phonons in the system. In the presence
of a laser emitting light at frequency
ω
L
, it is convenient to work in an interaction frame
where
ω
o
1
in the above Hamiltonian with
1
=
ω
o
ω
L
. To incorporate the effect of the
environment, we use the quantum-optical Langevin equations for the system [
31
33
],
̇
ˆ
b
(
t
)
=−
(
i
ω
m0
+
γ
i
2
)
ˆ
b
i
g
0
ˆ
a
ˆ
a
γ
i
ˆ
b
in
(
t
)
and
̇
ˆ
a
(
t
)
=−
(
i
1
+
κ
2
)
ˆ
a
i
g
0
ˆ
a
(
ˆ
b
+
ˆ
b
)
κ
e
/
2
ˆ
a
in
(
t
)
κ
ˆ
a
in
,
i
(
t
).
(2)
Here
ˆ
a
in
(
t
)
and
ˆ
a
in
,
i
(
t
)
are the quantum noise operators associated with extrinsic (input
/
output)
and intrinsic (undetected) optical loss channels, respectively. Here we assume bi-directional
evanescent waveguide coupling (see figure
1
) to the optical cavity in which the total extrinsic
cavity loss rate into both directions of the coupling waveguide channel is
κ
e
, with the uni-
directional input coupling rate being half that at
κ
e
/
2. The total optical cavity (energy) decay rate
is given by
κ
, with
κ
=
κ
κ
e
/
2 denoting all the optical loss channels which go undetected
6
.
The noise operator
ˆ
b
in
(
t
)
arises from the coupling of the mechanical system to the surrounding
bath degrees of freedom, which in most current systems resides in a high temperature thermal
state with average bath occupancy
n
b

1.
6
In an idealized measurement, all photons lost by the optical cavity are lost to the detected channel so that
κ
=
0.
In this ideal case, satisfied by the single-sided end-fire coupling geometry of figure
1
(c), the vacuum fluctuations
from ports other than the detector port never enter the optical cavity. For a bi-directional coupling scheme we have
κ
>
κ
e
/
2, and there is information lost in the backwards waveguide direction about the mechanical state.
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Figure 1.
(a) Bi-directional evanescent coupling geometry, in which the
‘transmitted’ field goes into the forward
a
out
waveguide channel and the
‘reflected’ field goes into the backward
a
out
waveguide channel. This is the
coupling geometry we will be focusing on in this work, in which the laser
input channel is
a
in
and the detection channel is the forward waveguide channel,
a
out
. (b) Double-sided end-fire coupling geometry. This is the geometry one
would have in a Fabry–Perot cavity. Note that in this geometry what we call
the ‘transmission’ and ‘reflection’ channels are typically opposite to those in
the evanescent coupling geometry (a direct map between the two geometries
would relate the ‘reflected’ channel in (a) to the conventional transmission of a
Fabry–Perot, for instance). (c) Single-sided end-fire coupling geometry. This is
the ideal measurement geometry, in which all of the optical signal that is coupled
into the cavity can be, in principle, collected and detected in the
a
out
channel. In
principle one does not have excess vacuum noise coupled into the optical cavity,
only that from the laser input channel
a
in
.
We linearize the equations about a large optical field intensity by displacing
ˆ
a
α
0
+
ˆ
a
.
Such an approximation is valid for systems such as ours where
g
0

κ
, and the optical vacuum
alone only marginally affects the dynamics of the system, i.e. the
vacuum weak coupling
regime.
All experimental systems to date are in this regime. For systems in the
vacuum strong coupling
(
g
0
) regime, more elaborate treatments taking into account the quantum nature of the
nonlinearity must be pursued [
42
44
]. In the Fourier domain the operators for the mechanical
and optical modes are found to be
ˆ
b
(ω)
=
γ
i
ˆ
b
in
(ω)
i
m0
ω)
+
γ
i
/
2
i
G
(
ˆ
a
(ω)
+
ˆ
a
(ω))
i
m0
ω)
+
γ
i
/
2
(3)
and
ˆ
a
(ω)
=
κ
e
/
2
ˆ
a
in
(ω)
κ
ˆ
a
in
,
i
i
G
(
ˆ
b
(ω)
+
ˆ
b
(ω))
i
(1
ω)
+
κ/
2
,
(4)
respectively, where
G
=
g
|
α
0
|
.
New Journal of Physics
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5
Using equations (
3
) and (
4
) we arrive at the operator for the mechanical fluctuations,
ˆ
b
(ω)
=
γ
i
ˆ
b
in
(ω)
i
m
ω)
+
γ/
2
+
i
G
i
(1
ω)
+
κ/
2
κ
e
/
2
ˆ
a
in
(ω)
+
κ
ˆ
a
in
,
i
(ω)
i
m
ω)
+
γ/
2
+
i
G
i
(1
+
ω)
+
κ/
2
κ
e
/
2
ˆ
a
in
(ω)
+
κ
ˆ
a
in
,
i
(ω)
i
m
ω)
+
γ/
2
,
(5)
where
ω
m
=
ω
m0
+
δω
m
is the optical spring shifted mechanical frequency and
γ
=
γ
i
+
γ
OM
is
the optically damped (or amplified) mechanical loss rate. Expressions for the optical springing
and damping terms are given by
δω
m
=|
G
|
2
Im
[
1
i
(1
ω
m
)
+
κ/
2
1
i
(1
+
ω
m
)
+
κ/
2
]
(6)
and
γ
OM
=
2
|
G
|
2
Re
[
1
i
(1
ω
m
)
+
κ/
2
1
i
(1
+
ω
m
)
+
κ/
2
]
,
(7)
respectively. From these expressions it is evident that in the sideband-resolved regime the
maximum optical damping occurs for a laser red-detuned from the optical cavity with
1
=
ω
m
,
resulting in a damping rate of
γ
OM
=
4
|
G
|
2
. The ratio between this optical contribution to the
mechanical damping and the intrinsic mechanical damping is the co-operativity,
C
γ
OM
i
.
2.1. Quantum-limited laser cooling and damping
The expression for the noise power spectrum of the laser driven mechanical system can be
calculated using equation (
5
) for
ˆ
b
(ω)
. More specifically we calculate
S
bb
(ω)
(see appendix
B
),
corresponding in the high-
Q
regime to the ability of the mechanical system to
emit
noise
power into its environment [
6
]. The area under
S
bb
(ω)
is the average mode occupancy of
the mechanical quantum oscillator. In the absence of the optical coupling to the mechanics
(
G
=
0), the result in appendix
B
is obtained. Allowing for optical coupling and including
the optical noise terms, we arrive at an expression involving the correlations
〈ˆ
a
in
(ω)
ˆ
a
in
)
,
〈ˆ
a
in
,
i
(ω)
ˆ
a
in
,
i
)
,
〈ˆ
a
in
(ω)
ˆ
a
in
)
and
〈ˆ
a
in
,
i
(ω)
ˆ
a
in
,
i
)
that must be calculated from the
properties of the optical bath. Assuming that our source of light is a pure coherent tone,
and thus the optical bath is in a vacuum state, as is approximately the case in many optical
experiments, the former two correlations can be set to zero, while the latter two give
δ(ω
+
ω
)
.
As described in appendix
B
, for the mechanical system which is in contact with a thermal bath
of occupancy
n
b
, we have noise input correlations of
ˆ
b
in
(ω)
ˆ
b
in
)
〉=
(
n
b
+ 1
)δ(ω
+
ω
)
and
ˆ
b
in
(ω)
ˆ
b
in
)
〉=
n
b
δ(ω
+
ω
)
. The expression for
S
bb
(ω)
is then found to be
S
bb
(ω)
=
γ
n
f
(ω)
m
+
ω)
2
+
(γ/
2
)
2
,
(8)
where
n
f
(ω)
, the back-action modified phonon occupation number, is given by
n
f
(ω)
=
γ
i
n
b
γ
+
|
G
|
2
κ
γ
1
(1
ω)
2
+
(κ/
2
)
2
.
(9)
In the
driven weak-coupling
regime

γ)
, the mechanical lineshape is not strongly modified
from that of a Lorentzian, and
n
f
(ω)
can simply be replaced by
n
f
(
ω
m
)
in equation (
8
).
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6
An input laser beam tuned a mechanical frequency red of the cavity for optimal laser cooling
(
1
=
ω
m
), results in a back-action modified average mechanical mode occupation number
equal to
〈ˆ
n
〉|
1
=
ω
m
=
γ
i
n
b
γ
+
γ
OM
γ
(
κ
4
ω
m
)
2
.
(10)
The term
n
qbl
(κ/
4
ω
m
)
2
is the quantum limit on back-action cooling, as derived in [
45
,
46
]
using master-equation methods and in [
47
] by taking into account the spectral density of the
optical back-action force. This small (in the good cavity limit) residual heating comes from the
non-resonant scattering of red pump photons, to one mechanical frequency lower, or a total of
2
ω
m
detuned from the optical cavity.
We note briefly that for other laser detunings different back-action occupancies are
achieved, such as
〈ˆ
n
〉|
1
=
0
=
n
b
+
4
|
G
|
2
γ
i
κ
(
κ
2
ω
m
)
2
(11)
and
〈ˆ
n
〉|
1
=−
ω
m
=
γ
i
n
b
γ
+
|
γ
OM
|
γ
,
(12)
where again the resolved-sideband limit is assumed, and for
1
=−
ω
m
one has amplification of
the mechanical motion with
γ
OM
=
4
|
G
|
2
(which must be smaller than
γ
i
to avoid triggering
instabilities.)
2.2. Thermometry with the cooling beam
One of the simplest methods of inferring the mechanical mode occupancy is to detect the
imprinted mechanical motion on the cooling laser beam itself. Upon transmission through
the cavity-optomechanical system, the laser cooling beam, typically detuned to
1
=
ω
m
in the
resolved sideband regime, preferentially picks up a blue-shifted sideband at frequency
ω
L
+
ω
m
(
ω
o
) due to removal of phonon quanta from the mechanical resonator (anti-Stokes scattering).
Upon detection with a photodetector, the beating of the anti-Stokes sideband with the intense
cooling tone produces an electrical signal at the mechanical frequency [
11
,
26
,
48
]. By careful
calibration and accurate measurement of the magnitude of this signal, and through independent
measurements of other system parameters such as
g
,
|
α
0
|
,
κ
e
, and
κ
, the mechanical resonator’s
average phonon number occupancy can be inferred.
The optical fluctuations in the transmitted laser cooling beam at the output port of the
optomechanical cavity are given approximately in the sideband resolved regime by
ˆ
a
out
(ω)
|
1
=
ω
m
t
;
1)
ˆ
a
in
(ω)
+
n
opt
;
1)
ˆ
a
in
,
i
(ω)
+
s
12
;
1)
ˆ
b
in
(ω),
(13)
where
t
,
n
opt
, and
s
12
are the scattering matrix elements evaluated for a laser cooling beam of red-
detuning
1
=
ω
m
(see appendix
D
). This expression is derived using equations (
3
) and (
4
) and
input–output boundary condition
ˆ
a
out
(ω)
= ˆ
a
in
(ω)
+
κ
e
/
2
ˆ
a
(ω)
. Expressions of this form have
been used previously to analyze the propagation of light and sound through an optomechanical
cavity in the context of state transfer [
49
,
50
]. In this case we have simplified equation (
13
) by
ignoring input noise terms from the creation operators
ˆ
a
in
(ω)
and
ˆ
a
in
,
i
(ω)
. These terms give rise
New Journal of Physics
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7
to the quantum-limit of laser cooling found in equation (
10
) above, but insignificantly modify
the optically transduced signal of the mechanical motion as long as
〈ˆ
n
〉
n
qbl
.
The strong cooling laser tone beats with the optical noise sidebands, generating a
photocurrent proportional to
ˆ
a
out
(
t
)
+
ˆ
a
out
(
t
)
,
ˆ
I
(ω)
|
1
=
ω
m
=
t
(ω)
ˆ
a
in
(ω)
+
n
opt
(ω)
ˆ
a
in
,
i
(ω)
+
s
12
(ω)
ˆ
b
in
(ω)
+ h
.
c
.(
ω),
where h
.
c
.(
ω)
is a convenient short-hand (
f
(ω)
+
(
f
(
ω))
=
f
(ω)
+ h
.
c
.(
ω)
). The
resulting photocurrent power spectral density as read out from a spectrum analyzer is
given by
S
II
(ω)
|
1
=
ω
m
=
−∞
d
ω
ˆ
I
(ω)
ˆ
I
)
=
−∞
t
(ω)
t
(
ω
)
〈ˆ
a
in
(ω)
ˆ
a
in
)
+
n
opt
(ω)
n
opt
(
ω
)
〈ˆ
a
in
,
i
(ω)
ˆ
a
in
,
i
)
+
s
12
(ω)
s
21
(
ω
)
ˆ
b
in
(ω)
ˆ
b
in
)
+
s
12
(
ω)
s
21
)
ˆ
b
in
(ω)
ˆ
b
in
)
d
ω
,
(14)
where we have assumed the same optical (vacuum) and mechanical (thermal) noise correlations
as above in evaluating equation (
8
). Using the normalization property of the scattering matrix
coefficients (
|
t
(ω)
|
2
+
|
n
opt
(ω)
|
2
+
|
s
12
(ω)
|
2
=
1), we find the simplified expression
S
II
(ω)
|
1
=
ω
m
=
1 +
n
b
(
|
s
12
(ω)
|
2
+
|
s
12
(
ω)
|
2
).
(15)
Substituting for the expression of the phonon–photon scattering element
s
12
(ω)
given in
appendix
D
for
1
=
ω
m
yields
S
II
(ω)
|
1
=
ω
m
=
1 +
κ
e
2
κ
8
|
G
|
2
κ
̄
S
bb
;〈ˆ
n
),
(16)
for the transduced noise power spectral density, where
〈ˆ
n
is the actual mechanical mode
occupancy including back-action effects of the cooling laser (see equation (
10
)).
Several points are worth mentioning regarding this expression. Firstly, the signal to noise
goes as the coupling efficiency
η
=
κ
e
/
2
κ
. In the case of non-ideal measurement, i.e. losses
in the optical path from cavity to detector, or an imperfect detector, the efficiency would be
multiplied by an additional factor
η
detection
which is taken to be unity throughout this work (this
point is discussed further in the experimental results section). Secondly, the detected signal
is proportional to
ˆ
b
ˆ
b
as opposed to
ˆ
b
ˆ
b
+ 1
/
2, and so the resulting signal is exactly what
would be expected classically, vanishing as the temperature and phonon occupation go to zero.
In other words, this measurement is insensitive to the zero-point motion of the resonator. The
spectral density
S
bb
;〈ˆ
n
)
, represents the ability of the mechanical system to
emit
energy [
6
].
By tuning the laser to
1
=
ω
m
in the sideband-resolved regime it is exceedingly unlikely for the
tone to drive the mechanics (since the required Stokes scattering process becomes non-resonant
by 2
ω
m
), and so we gain little information about how the mechanical system
absorbs
energy
from the optical bath. Finally, we note that equation (
16
) is general and holds for both low and
high cooperativity.
2.2.1. Interpretation as quantum noise squashing.
Though the scattering matrix formulation
provides a consistent and systematic way of deriving the form of the detected signals, it does
so by elimination of the position operator from the equations. It is interesting to reinterpret the
experiment as a measurement of the position of the mechanical system [
6
], and we attempt to
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8
do so here. A much more thorough treatment of the implications of quantum back-action and
measurement theory in this type of system is presented in a recent work by Khalili
et al
[
51
].
For simplicity, the perfect coupling condition is assumed, i.e.
κ
e
/
2
=
κ
. The output signal is
then given by
ˆ
a
out
(ω)
|
1
=
ω
m
≈−ˆ
a
in
(ω)
i2
G
κ
ˆ
b
(ω).
The normalized homodyne current is found to be
ˆ
I
(
t
)
|
1
=
ω
m
=−
i
ˆ
a
in
(
t
)
+ i
ˆ
a
in
(
t
)
+
2
G
κ
(
ˆ
b
(
t
)
+
ˆ
b
(
t
))
(17)
and so it would seem that the signal
ˆ
I
(
t
)
is composed of optical shot-noise and a component
which is proportional to
ˆ
x
, making
S
II
=
1 + const
×〈ˆ
x
2
. This however contradicts the above
derivations which show that
S
II
=
1 + const
×〈ˆ
n
for
1
=
ω
m
. The inconsistency comes
after careful calculation of the correlation function
ˆ
I
(
t
+
τ)
ˆ
I
(
t
)
. In fact,
ˆ
a
in
(
t
)
and
ˆ
b
(
t
)
are correlated, and the view that the shot-noise simply creates a constant noise floor is
incorrect [
52
,
53
]. Proper accounting for the correlations (see appendix
C
) leads us again to
equation (
16
), showing that the measured quantity is
̄
S
bb
, and the area of the detected spectrum
is proportional to
〈ˆ
n
.
The blue-side driving with
1
=−
ω
m
causes the opposite effect, i.e. quantum noise anti-
squashing. The squashing and anti-squashing are signatures of quantum back-action. This
effect is in spirit similar to classical noise squashing which we study in section
2.4.2
, where
correlations between the noise-induced motion and
classical
noise of the detection beam
destructively interfere at the photodetector. It is important to note that this signature of quantum
back-action does not involve detection of quantum back-action heating, and can be apparent at
arbitrarily low powers, far below that required to reach the standard quantum limit.
2.3. Motional sideband asymmetry thermometry
An alternate method of measuring the temperature of the mechanical subsystem, one which uses
the mechanical zero-point motion to self-calibrate the measured phonon occupancy, involves
comparing the measured signal from a weak probe beam (low cooperativity) at both
1
ω
m
in the sideband resolved regime [
27
]. In such experiments, the mechanics can be either laser
cooled with a different laser and
/
or optical cavity mode, or the system can be cryogenically pre-
cooled to a temperature which requires no further cooling to approach the quantum ground state.
As the optical read-out beam can be arbitrarily weak in such measurements, it only marginally
affects the dynamics of the mechanical system [
14
,
27
,
54
]. By working at low read-out
beam power, such that the optically-induced damping and amplification rates are much smaller
than the bare mechanical linewidth, optical back-action by the probe beam only minimally
affects the dynamics of the mechanical system and measurements can be taken at detunings
both red (
1
=
ω
m
) and blue (
1
=−
ω
m
) of the cavity without triggering any optomechanical
instabilities [
55
]. Operating in the resolved sideband regime allows for the separate cavity
filtering of the Stokes and anti-Stokes motionally induced sidebands on the probe beam, which
are respectively proportional to
〈ˆ
n
+ 1 and
〈ˆ
n
. It can be shown that the additional vacuum
contribution to the Stokes scattering, which provides the intrinsic calibration for
〈ˆ
n
, arises in
these measurements equally from the shot noise on the probe laser and zero-point motion of the
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9
mechanical resonator [
51
]. We will also see in the following sections, that such a measurement
at both
1
ω
m
can provide additional resilience to systematic errors from non-idealities such
as laser phase noise.
We derive here the blue-detuned (
1
=−
ω
m
) result analogous to the red-detuned (
1
=
ω
m
) laser cooling case given above in equation (
16
)). In the sideband-resolved regime, the
approximations that led to equation (
13
), lead to a similar expression in the case of
1
=−
ω
m
for the electromagnetic field output from the optomechanical cavity
ˆ
a
out
(ω)
|
1
=−
ω
m
t
;
1)
ˆ
a
in
(ω)
+
n
opt
;
1)
ˆ
a
in
,
i
(ω)
+
s
12
;
1)
ˆ
b
in
(ω),
(18)
where we have neglected the terms proportional to the photon noise creation operators as their
effect is minimal on the optically transduced signal of the mechanical motion in the sideband-
resolved weak coupling regime when
n
〉
n
qbl
. The scattering relation, whose exact form is
shown in the appendix
D
, allows the optomechanical system to act as an amplifier, and has
been studied experimentally at microwave [
20
] and optical frequencies [
56
], and studied more
generally in the context of optomechanics by Botter
et al
[
57
]. The scattering elements satisfy
the equation
|
t
(ω)
|
2
+
|
n
opt
(ω)
|
2
+
|
s
12
(ω)
|
2
=
1, which along with the standard bath correlation
relations used above, allows us to write
S
II
(ω)
|
1
=−
ω
m
=
1 +
(
n
b
+ 1
)(
|
s
12
;
1)
|
2
+
|
s
12
(
ω
;
1)
|
2
)
(19)
=
1 +
κ
e
2
κ
8
|
G
|
2
κ
̄
S
b
b
;〈ˆ
n
),
(20)
where
〈ˆ
n
is the actual mode occupancy including back-action of the laser input (see
equation (
12
)). As before, the signal lies on top of a flat shot noise background of unity, and
is proportional to the detection efficiency
η
and the measurement rate
γ
OM
. Now, however,
the signal is proportional to the creation operator spectral density
̄
S
b
b
;〈ˆ
n
)
, which itself is
proportional to
〈ˆ
n
+ 1. The spectral density
̄
S
b
b
;〈ˆ
n
)
can be interpreted as the mechanical
system’s ability to
absorb
energy [
5
], which even at zero temperature (occupation) can absorb
energy through a spontaneous scattering process which arises due to the zero-point motion of
the mechanical resonator.
For a constant laser driving power the optomechanical damping and amplification rates for
detuning
1
ω
m
are equal in magnitude but opposite in sign, with
γ
±
γ
i
±|
γ
OM
|
, where
|
γ
OM
|
=
4
|
G
|
2
in the sideband resolved, weak coupling regime. Weak probing entails using
a probe intensity such that
|
γ
OM
|
γ
i
, or
C
r

1, where we define
C
r
≡|
γ
OM
|
i
as the read-
out beam cooperativity. In this limit the mechanical mode occupation numbers for
1
ω
m
detunings are given approximately by
〈ˆ
n
±
=
γ
i
n
b
±
, where
n
b
is the mechanical mode
occupancy in absence of the probe field
7
. Denoting the integrated area under the Lorentzians
in equations (
16
) and (
20
) as
I
+
and
I
, respectively, we find a relation between their ratios
and the read-out cooperativity which provides a quantum calibration of the unperturbed thermal
occupancy [
27
]:
η
I
/
I
+
1 +
C
r
1
1
C
r
=
1
n
b
.
(21)
7
Referring to equations (
10
) and (
12
), this is an accurate relation if
C
r

n
b
.
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2.4. Laser phase noise
Although various other noise sources (laser intensity noise, internal cavity noise [
18
], etc) can be
treated similarly, here we focus on laser phase noise as it is typically the most important source
of non-ideality in the laser cooling and thermometry of cavity opto-mechanical systems. In our
measurements, we have ruled out intensity noise as a significant source of systematic error, as
described in section
3.2
. Detection of laser phase noise and its effect on the experiment is more
involved, and recent measurements of phase noise in lasers of a similar make and model as ours
have been reported [
58
]. The effect of phase noise on optomechanical systems has also been of
great interest and studied at depth in the context of heating [
39
] and entanglement [
40
,
59
,
60
]
of light and mechanics. However, laser light often acts as both a means by which the mechanical
system is cooled as well as its temperature measured, and thus laser noise can affect both the
true and inferred mechanical mode occupancy. Here we complement previous studies of laser
noise heating with a unified analysis that also quantifies the effects of quantum and classical
(phase) laser noise on the optically-transduced mechanical mode spectra.
The optical laser field amplitude input to the optomechanical system, in a rotating reference
frame at frequency
ω
L
and in units of
photons s
1
, we denote by
E
0
. Due to processes internal
to the laser, some fundamental in nature, others technical, this amplitude undergoes random
phase fluctuations which are captured by adding a random rotating phase factor [
61
]
E
0
(
t
)
=|
E
0
|
e
i
φ(
t
)
.
(22)
As long as the phase fluctuations are small, we expand this expression to first order yielding
E
0
(
t
)
≈|
E
0
|
(
1 + i
φ(
t
)
+
O
2
))
[
39
]. Then
E
0
(τ)
E
0
(
0
)
〉=|
E
0
|
2
(
1 +
φ(τ)φ(
0
)
)
.
In this way, we can express the noise power spectral density of the optical field amplitude,
S
E E
(ω)
, as
S
E E
(ω)
=|
E
0
|
2
(
2
πδ(ω)
+
̄
S
φφ
(ω)),
(23)
where we have also used the realness of
φ(
t
)
to set
S
φφ
(ω)
=
̄
S
φφ
(ω)
. The
δ(ω)
term is due to
the carrier. We are interested in fluctuations of laser light away from the carrier and so we will
ignore this term from here onward.
This relates the phase noise power spectral density to the optical power spectrum of the
noisy laser beam, with the optical power away from the carrier at
ω
=
0 due to phase noise.
This phase noise can then be taken into account as an additional noise input to the cavity,
ˆ
a
in
,
tot
(ω)
= ˆ
a
in
(ω)
+
a
in
(ω),
(24)
where
a
in
(ω)
is a stochastic input with
a
in
(ω)
a
in
)
〉=
S
E E
(ω)δ(ω
+
ω
)
(here the
averages used for correlation functions correspond to classical ensemble averages and
a
in
(ω)
(
a
in
(
ω))
).
There is, however, an additional subtlety when performing mechanical mode thermometry
with a laser beam affected by phase noise; correlations between the positive and negative
frequency components of the phase noise can cause cancellations in the optically transduced
signal, and therefore must be carefully taken into account. For example, for a pure sinusoidal
tone phase modulated onto a laser we have
E
0
e
i
φ(
t
)
'
E
0
(
e
i
β
c
cos
ω
t
+i
β
s
sin
ω
t
)
'
E
0
(
1 +
1
2
s
+ i
β
c
)
e
i
ω
t
1
2
s
i
β
c
)
e
i
ω
t
)
.
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11
The positive and negative frequency optical sideband amplitudes are negative complex
conjugates of one another. More generally, the positive and negative frequency components
of the noisy optical input have the following relation for an optical signal with phase noise:
a
(
)
in
(ω)
=−
(
a
(
+
)
in
(
ω)
)
,
(25)
where
a
(
)
in
(ω)
=
θ(
ω)
a
in
(ω)
, and
a
(
+
)
in
(ω)
=
θ(ω)
a
in
(ω)
(with
θ(ω)
=
1 for
ω>
0,
θ(ω)
=
0
for
ω <
0). The total phase noise signal can then be expressed as
a
in
(ω)
=
a
(
+
)
in
(ω)
a
(
+
)
in
(
ω)
. For calculations that follow, this explicit separation of positive and negative
frequency phase noise components is useful in simplifying calculations of the optically
transduced mechanical motion. In terms of positive frequency phase noise components only
then, we have
a
(
+
)
in
(ω)
a
(
+
)
in
)
〉=
S
E E
(ω)δ(ω
+
ω
)θ(ω),
(26)
with a similar relation holding for the negative frequency components of the phase noise input
and the negative frequency optical power spectrum.
2.4.1. Heating.
To find the actual thermal occupation of the mechanical system in the presence
of phase noise on the laser cooling beam we use once again equation (
5
) for
ˆ
b
(ω)
, replacing
ˆ
a
in
(ω)
with
ˆ
a
in
,
tot
(ω)
which includes the classical phase noise on the input laser. From the non-
zero correlation
〈ˆ
a
in
,
tot
(ω)
ˆ
a
in
,
tot
)
〉=
S
E E
(ω)δ(ω
+
ω
)
for
ω,ω
>
0, we find another source
of noise phonons, in addition to those coming from the thermal bath and quantum back-action
of the laser light. This is expressed as new terms proportional to
S
E E
(ω)
in the noise spectrum
of the mechanical motion given by equation (
8
),
S
bb
(ω)
=
γ
n
f
(ω)
m
+
ω)
2
+
(γ/
2
)
2
,
(27)
where
n
f
(ω)
|
1
=
ω
m
=
γ
i
n
b
γ
+
|
G
|
2
κ
γ
1 +
e
/
2
κ)
S
E E
(ω)
(1
ω)
2
+
(κ/
2
)
2
+
|
G
|
2
κ
γ
e
/
2
κ)
S
E E
(ω)
(1
+
ω)
2
+
(κ/
2
)
2
.
(28)
As before, assuming a high mechanical
Q
-factor and the driven weak-coupling regime

γ)
,
we substitute
n
f
(
ω
m
)
for
n
f
(ω)
and relate it to the average mode occupancy in the presence
of laser phase noise,
〈ˆ
n
φ
,
〈ˆ
n
φ
|
1
=
ω
m
=
γ
i
n
b
γ
+
γ
OM
γ
[
(
κ
4
ω
m
)
2
+
(
κ
e
2
κ
)
n
φ
]
,
(29)
where we have defined
n
φ
S
E E
m
)
[
39
].
The additional phase noise heating in equation (
29
) can be understood as the product of the
number of noise photons present in the light field at a mechanical frequency detuned from
the central laser frequency (
n
φ
), multiplied by the efficiency with which they are coupled
into the cavity (
κ
e
/
2
κ
), and finally multiplied by the efficiency with which the light field couples
to the mechanics (
γ
OM
). For detuning
1
=
ω
m
used in resolved sideband laser cooling, the
optomechanical system only samples the input laser phase noise at a mechanical frequency
blue of the central laser frequency, and thus the relationship between the negative and positive
frequency components of the laser phase noise has no role to play in heating in the sideband-
resolved regime (as we will see below, this is not the case in the thermometry).
New Journal of Physics
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