of 36
Laser noise in cavity-optomechanical cooling and
thermometry
Amir H. Safavi-Naeini
1
, Jasper Chan
1
, Jeff T. Hill
1
, Simon
Gr ̈oblacher
1
,
2
, Haixing Miao
3
, Yanbei Chen
3
, Markus
Aspelmeyer
4
, Oskar Painter
1
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 Blvd., 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, opainter@caltech.edu
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
À
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.
arXiv:1210.2671v1 [physics.optics] 9 Oct 2012
Laser noise in cavity-optomechanical cooling and thermometry
2
1. Introduction
Mechanical resonators provide an interesting and useful system for the study of quantum
theory in the mesoscopic and macroscopic scales [1, 2, 3, 4, 5, 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, 8, 9, 10, 11, 12, 13], microwave resonators [14, 15, 16, 17],
superconducting qubits [18, 19, 20], cold atoms [21], and defect centers in diamond [22]
have been forthcoming. These advances have led to the cooling of mechanical systems
to their quantum ground states [19, 16, 23], and the observation of nonclassical
behaviour [19, 24, 25]. 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, to cool optomechanical systems
close to their ground states using radiation pressure forces, and recently experiments
starting with modest cryogenic pre-cooling [11, 23, 24, 26], 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 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 [23, 24], i.e. the driven weak-coupling, sideband-resolved regime where the
cavity decay rate is larger than all other rates in the system except for the mechanical
frequency. In this regime, the standard input-ouput formalism [27, 28, 29] for analyzing
the linearized system is applied. Two different methods of thermometry used in recent
experiments [23, 24] 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 subsystem, and observing
a non-classical asymmetry analogous to that seen in much earlier experiments with
trapped atoms and ions [30, 31, 32]. Both methods are susceptible to laser phase noise,
which is discussed in section 2.4, through both noise induced heating [33], and systematic
errors in thermometry caused by noise
squashing
[34, 15] and
anti-squashing
. Finally,
in Section 3 we review recent laser cooling and sideband asymmetry measurements
of optomechanical crystal 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 experiements.
Laser noise in cavity-optomechanical cooling and thermometry
3
2. Theory
a
b
a
in
a’
out
a
out
a’
in
°
i
·
e
/2
·
-
·
e
g
o
·
e
/2
a
b
a
in
a
out
°
i
·
-
·
e
g
o
(a)
(b)
a
b
a
in
a
out
°
i
·
e
·
-
·
e
g
o
(c)
a’
out
·
e
/2
·
e
/2
a’
in
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
1
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 of that 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 couplng 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
.
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
~
ω
o
ˆ
a
:
ˆ
a
`
~
ω
m
0
ˆ
b
:
ˆ
b
`
~
g
ˆ
a
:
ˆ
a
p
ˆ
b
:
`
ˆ
b
q
,
(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
Ñ
∆ in the above Hamiltonian with ∆
ω
o
́
ω
L
.
To incorporate the effect of the environment, we use the quantum-optical Langevin
equations for the system [27, 28, 29],
9
ˆ
b
p
t
q “ ́
́
m
0
`
γ
i
2
̄
ˆ
b
́
ig
ˆ
a
:
ˆ
a
́
?
γ
i
ˆ
b
in
p
t
q
and
9
ˆ
a
p
t
q “ ́
́
i
`
κ
2
̄
ˆ
a
́
ig
ˆ
a
p
ˆ
b
:
`
ˆ
b
q ́
a
κ
e
{
a
in
p
t
q ́
?
κ
1
ˆ
a
in
,
i
p
t
q
.
(2)
Here ˆ
a
in
p
t
q
and ˆ
a
in
,
i
p
t
q
are the quantum noise operators associated with extrinsic
(input/output) and intrinsic (undetected) optical loss channels, respectively. Here we
Laser noise in cavity-optomechanical cooling and thermometry
4
assume bi-directional evanescent waveguide coupling (see Fig. 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
1
κ
́
κ
e
{
2 denoting all the
optical loss channels which go undetected.
;
The noise operator
ˆ
b
in
p
t
q
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.
We linearize the equations about a large optical field intensity by displacing
ˆ
a
Ñ
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
ą
κ
) regime, more elaborate treatments
taking into account the quantum nature of the nonlinearity must be pursued [35, 36, 37].
In the Fourier domain the operators for the mechanical and optical modes are found to
be
ˆ
b
p
ω
q “
́
?
γ
i
ˆ
b
in
p
ω
q
i
p
ω
m
0
́
ω
q`
γ
i
{
2
́
iG
p
ˆ
a
p
ω
q`
ˆ
a
:
p
ω
qq
i
p
ω
m
0
́
ω
q`
γ
i
{
2
(3)
and
ˆ
a
p
ω
q “
́
a
κ
e
{
a
in
p
ω
q ́
?
κ
1
ˆ
a
in
,
i
́
iG
p
ˆ
b
p
ω
q`
ˆ
b
:
p
ω
qq
i
p
́
ω
q`
κ
{
2
,
(4)
respectively, where
G
g
|
α
0
|
.
Using equations (3-4) we arrive at the operator for the mechanical fluctuations,
ˆ
b
p
ω
q “
́
?
γ
i
ˆ
b
in
p
ω
q
i
p
ω
m
́
ω
q`
γ
{
2
`
iG
i
p
́
ω
q`
κ
{
2
a
κ
e
{
a
in
p
ω
q`
?
κ
1
ˆ
a
in
,
i
p
ω
q
i
p
ω
m
́
ω
q`
γ
{
2
`
iG
́
i
p
`
ω
q`
κ
{
2
a
κ
e
{
a
:
in
p
ω
q`
?
κ
1
ˆ
a
:
in
,
i
p
ω
q
i
p
ω
m
́
ω
q`
γ
{
2
,
(5)
where
ω
m
ω
m
0
`
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
p
́
ω
m
q`
κ
{
2
́
1
́
i
p
`
ω
m
q`
κ
{
2
(6)
and
γ
OM
2
|
G
|
2
Re
1
i
p
́
ω
m
q`
κ
{
2
́
1
́
i
p
`
ω
m
q`
κ
{
2
,
(7)
;
In an idealized measurement, all photons lost by the optical cavity are lost to the detected channel
so that
κ
1
0. In this ideal case, satisfied by the single-sided end-fire coupling geometry of Fig. 1c,
the vacuum fluctuations from ports other than the detector port never enter the optical cavity. For
a bi-directional coupling scheme we have
κ
1
ě
κ
e
{
2, and there is information lost in the backwards
waveguide direction about the mechanical state.
Laser noise in cavity-optomechanical cooling and thermometry
5
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 ∆
ω
m
, resulting in a damping rate of
γ
OM
4
|
G
|
2
{
κ
. The ratio between this
optical contribution to the mechanical damping and the intrinsic mechanical damping
is the co-operativity,
C
γ
OM
{
γ
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 eqn. (5) for
ˆ
b
p
ω
q
. More specifically we calculate
S
bb
p
ω
q
(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
p
ω
q
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
x
ˆ
a
:
in
p
ω
q
ˆ
a
in
p
ω
1
qy
,
x
ˆ
a
:
in
,
i
p
ω
q
ˆ
a
in
,
i
p
ω
1
qy
,
x
ˆ
a
in
p
ω
q
ˆ
a
:
in
p
ω
1
qy
and
x
ˆ
a
in
,
i
p
ω
q
ˆ
a
:
in
,
i
p
ω
1
qy
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
p
ω
`
ω
1
q
. 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
x
ˆ
b
in
p
ω
q
ˆ
b
:
in
p
ω
1
qy “ p
n
b
`
1
q
δ
p
ω
`
ω
1
q
and
x
ˆ
b
:
in
p
ω
q
ˆ
b
in
p
ω
1
qy “
n
b
δ
p
ω
`
ω
1
q
. The expression for
S
bb
p
ω
q
is then found to be,
S
bb
p
ω
q “
γn
f
p
ω
q
p
ω
m
`
ω
q
2
`p
γ
{
2
q
2
,
(8)
where
n
f
p
ω
q
, the back-action modified phonon occupation number, is given by
n
f
p
ω
q “
γ
i
n
b
γ
`
|
G
|
2
κ
γ
1
p
́
ω
q
2
`p
κ
{
2
q
2
.
(9)
In the
driven weak-coupling
regime
p
κ
"
γ
q
, the mechanical lineshape is not strongly
modified from that of a Lorentzian, and
n
f
p
ω
q
can simply be replaced by
n
f
p ́
ω
m
q
in eqn. (8). An input laser beam tuned a mechanical frequency red of the cavity for
optimal laser cooling (∆
ω
m
), results in a back-action modified average mechanical
mode occupation number equal to
x
ˆ
n
y
|
ω
m
γ
i
n
b
γ
`
γ
OM
γ
ˆ
κ
4
ω
m
̇
2
.
(10)
The term
n
qbl
” p
κ
{
4
ω
m
q
2
is the quantum limit on back-action cooling, as derived in
[38, 39] using master-equation methods and in [40] 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.
Laser noise in cavity-optomechanical cooling and thermometry
6
We note briefly that for other laser detunings different back-action occupancies are
achieved, such as
x
ˆ
n
y
|
0
n
b
`
4
|
G
|
2
γ
i
κ
ˆ
κ
2
ω
m
̇
2
(11)
and
x
ˆ
n
y
|
“ ́
ω
m
γ
i
n
b
γ
`
|
γ
OM
|
γ
,
(12)
where again the resolved-sideband limit is assumed, and for ∆
“ ́
ω
m
one has
amplification of the mechanical motion with
γ
OM
– ́
4
|
G
|
2
{
κ
.
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
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 [10, 41, 23]. 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
p
ω
q
|
ω
m
«
t
p
ω
; ∆
q
ˆ
a
in
p
ω
q`
n
opt
p
ω
; ∆
q
ˆ
a
in
,
i
p
ω
q`
s
12
p
ω
; ∆
q
ˆ
b
in
p
ω
q
,
(13)
where
t
,
n
opt
, and
s
12
are the scattering matrix elements evaluated for a laser cooling
beam of red-detuning ∆
ω
m
(see Appendix D). This expression is derived using eqns.
(3-4) and input-output boundary condition ˆ
a
out
p
ω
q “
ˆ
a
in
p
ω
q`
a
κ
e
{
a
p
ω
q
. 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 [42, 43]. In this case
we have simplified eqn. (13) by ignoring input noise terms from the creation operators
ˆ
a
:
in
p
ω
q
and ˆ
a
:
in
,
i
p
ω
q
. These terms gives rise to the quantum-limit of laser cooling found
in eqn. (10) above, but insignificantly modify the optically transduced signal of the
mechanical motion as long as
x
ˆ
n
y "
n
qbl
.
The strong cooling laser tone beats with the optical noise sidebands, generating a
photocurrent proportional to ˆ
a
out
p
t
q`
ˆ
a
:
out
p
t
q
,
ˆ
I
p
ω
q
|
ω
m
t
p
ω
q
ˆ
a
in
p
ω
q`
n
opt
p
ω
q
ˆ
a
in
,
i
p
ω
q`
s
12
p
ω
q
ˆ
b
in
p
ω
q`
`
h
.
c
.
p ́
ω
q
Laser noise in cavity-optomechanical cooling and thermometry
7
where h
.
c
.
p ́
ω
q
is a convenient short-hand (
f
p
ω
q ` p
f
p ́
ω
qq
:
f
p
ω
q `
h
.
c
.
p ́
ω
q
). The
resulting photocurrent power spectral density as read out from a spectrum analyzer is
given by
S
II
p
ω
q
|
ω
m
ż
8
́8
d
ω
1
x
ˆ
I
:
p
ω
q
ˆ
I
p
ω
1
qy
ż
8
́8
t
p
ω
q
t
p ́
ω
1
q
̊
x
ˆ
a
in
p
ω
q
ˆ
a
:
in
p
ω
1
qy
`
n
opt
p
ω
q
n
opt
p ́
ω
1
q
̊
x
ˆ
a
in
,
i
p
ω
q
ˆ
a
:
in
,
i
p
ω
1
qy
`
s
12
p
ω
q
s
21
p ́
ω
1
q
̊
x
ˆ
b
in
p
ω
q
ˆ
b
:
in
p
ω
1
qy
`
s
12
p ́
ω
q
s
21
p
ω
1
q
̊
x
ˆ
b
:
in
p
ω
q
ˆ
b
in
p
ω
1
qy
d
ω
1
,
(14)
where we have assumed the same optical (vacuum) and mechanical (thermal) noise
correlations as above in evaluating eqn. (8). Using the normalization property of the
scattering matrix coefficients (
|
t
p
ω
q|
2
`|
n
opt
p
ω
q|
2
`|
s
12
p
ω
q|
2
1), we find the simplified
expression
S
II
p
ω
q
|
ω
m
1
`
n
b
p|
s
12
p
ω
q|
2
`|
s
12
p ́
ω
q|
2
q
.
(15)
Substituting for the expression of the phonon-photon scattering element
s
12
p
ω
q
given in
Appendix D for ∆
ω
m
yields
S
II
p
ω
q
|
ω
m
1
`
κ
e
2
κ
8
|
G
|
2
κ
̄
S
bb
p
ω
;
x
ˆ
n
yq
,
(16)
for the transduced noise power spectral density, where
x
ˆ
n
y
is the actual mechanical
mode occupancy including back-action effects of the cooling laser (see eqn. (10)).
Several points are worth mentioning regarding this expression. Firstly, the signal
to noise goes as the coupling efficiency
η
κ
e
{
2
κ
. Secondly, the detected signal is
proportional to
x
ˆ
b
:
ˆ
b
y
as opposed to
x
ˆ
b
:
ˆ
b
y `
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
p
ω,
x
ˆ
n
yq
, represents the ability of the mechanical
system to
emit
energy [6]. By tuning the laser to ∆
ω
m
in the sideband-resolved
regime it is exceedingly unlikely for the tone to drive the mechanics (through Stokes
scattering), 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.
Intepretation 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 do so here
§
. For simplicity, the perfect
§
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. [44]
Laser noise in cavity-optomechanical cooling and thermometry
8
coupling condition is assumed, i.e.
κ
e
{
2
κ
. The output signal is then given by
ˆ
a
out
p
ω
q
|
ω
m
« ́
ˆ
a
in
p
ω
q ́
i
2
G
?
κ
ˆ
b
p
ω
q
The normalized heterodyne current is found to be
ˆ
I
p
t
q
|
ω
m
“ ́
i
ˆ
a
in
p
t
q`
i
ˆ
a
:
in
p
t
q`
2
G
?
κ
p
ˆ
b
p
t
q`
ˆ
b
:
p
t
qq
(17)
and so it would seem that the signal
ˆ
I
p
t
q
is composed of optical shot-noise and a
component which is proportional to ˆ
x
, making
S
II
1
`
const
ˆ x
ˆ
x
2
y
. This however
contradicts the above derivations which show that
S
II
1
`
const
ˆx
ˆ
n
y
for ∆
ω
m
. The
inconsistency comes after careful calculation of the correlation function
x
ˆ
I
p
t
`
τ
q
ˆ
I
p
t
qy
.
In fact, ˆ
a
:
in
p
t
q
and
ˆ
b
p
t
q
are correlated, and the view that the shot-noise simply creates a
constant noise floor is incorrect. Proper accounting for the correlations (see Appendix
C) leads us again to eqn. (16), showing that the measured quantity is
̄
S
bb
, and the area
of the detected spectrum is proportional to
x
ˆ
n
y
.
The blue-side driving with ∆
“ ́
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 arbitrarly 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 ∆
“ ̆
ω
m
in the sideband resolved regime [24].
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 [45, 11, 24]. 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 (∆
ω
m
) and blue (∆
“ ́
ω
m
) of the cavity
without triggering any optomechanical instabilities [46]. 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
x
ˆ
n
y`
1 and
x
ˆ
n
y
. It can be shown that the additional vacuum contribution to the Stokes
Laser noise in cavity-optomechanical cooling and thermometry
9
scattering, which provides the intrinsic calibration for
x
ˆ
n
y
, arises in these measurements
equally from the shot noise on the probe laser and zero-point motion of the mechanical
resonator [44]. We will also see in the following sections, that such a measurement at
both ∆
“ ̆
ω
m
can provide additional resilience to systematic errors from non-idealities
such as laser phase noise.
We derive here the blue-detuned (∆
“ ́
ω
m
) result analogous to the red-detuned
(∆
ω
m
) laser cooling case given above in eqn.(16)). In the sideband-resolved regime,
the approximations that led to eqn. (13), lead to a similar expression in the case of
“ ́
ω
m
for the electromagnetic field output from the optomechanical cavity
ˆ
a
out
p
ω
q
|
“ ́
ω
m
«
t
p
ω
; ∆
q
ˆ
a
in
p
ω
q`
n
opt
p
ω
; ∆
q
ˆ
a
in
,
i
p
ω
q`
s
12
p
ω
; ∆
q
ˆ
b
:
in
p
ω
q
.
(18)
where we have neglected the terms proportional to the photon noise creation operators as
their effect is again minimal on the optically transduced signal of the mechanical motion.
Such a 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 [17] and optical frequencies [47], and studied more generally in the context
of optomechanics by Botter, et al. [48]. The scattering elements satisfy the equation
|
t
p
ω
q|
2
` |
n
opt
p
ω
q|
2
` |
s
12
p
ω
q|
2
1, which along with the standard bath correlation
relations used above, allows us to write
S
II
p
ω
q
|
“ ́
ω
m
1
`p
n
b
`
1
qp|
s
12
p
ω
; ∆
q|
2
`|
s
12
p ́
ω
; ∆
q|
2
q
(19)
1
`
κ
e
2
κ
8
|
G
|
2
κ
̄
S
b
:
b
:
p
ω
;
x
ˆ
n
yq
,
(20)
where
x
ˆ
n
y
is the actual mode occupancy including back-action of the laser input (see
eqn. (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
:
p
ω
;
x
ˆ
n
yq
, which
itself is proportional to
x
ˆ
n
y `
1. The spectral density
̄
S
b
:
b
:
p
ω
;
x
ˆ
n
yq
can be interpreted
as the mechanical system’s ability to
absorb
energy [5], which even at zero temperature
(occupation) can absorb energy through 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 ∆
“ ̆
ω
m
are equal in magnitude but opposite in sign, with
γ
̆
γ
i
̆ |
γ
OM
|
, where
|
γ
OM
| –
4
|
G
|
2
{
κ
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
|{
γ
i
as the read-out beam cooperativity. In this limit the
mechanical mode occupation numbers for ∆
“ ̆
ω
m
detunings are given approximately
by
x
ˆ
n
y
̆
γ
i
n
b
{
γ
̆
, where
n
b
is the mechanical mode occupancy in absence of the
probe field
}
. Denoting the integrated area under the Lorentzians in eqns. (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
}
Referring to eqns. (10) and (12), this is an accurate relation if
C
r
!
n
b
.
Laser noise in cavity-optomechanical cooling and thermometry
10
occupancy [24]:
η
I
́
{
I
`
1
`
C
r
́
1
1
́
C
r
1
n
b
.
(21)
2.4. Laser phase noise
Although various other noise sources (laser intensity noise, internal cavity noise [15], etc.)
can be treated similarly, here we focus on laser phase noise as it is typically the most
important source of nonideality in the laser cooling and thermometry of cavity opto-
mechanical systems. The effect of phase noise on optomechanical systems has already
been studied at great depth in the context of heating [33] of the mechanical resonator
and entanglement [49, 50, 51] 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 effect 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
a
photons/s, 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 [52]
E
0
p
t
q “ |
E
0
|
e
p
t
q
.
(22)
As long as the phase fluctuations are small, we expand this expression to first order
yielding
E
0
p
t
q « |
E
0
|p
1
`
p
t
q`
O
p
φ
2
qq
[33]. Then
x
E
̊
0
p
τ
q
E
0
p
0
qy “ |
E
0
|
2
p
1
`x
φ
p
τ
q
φ
p
0
qyq
.
In this way, we can express the noise power spectral density of the optical field amplitude,
S
EE
p
ω
q
, as
S
EE
p
ω
q “ |
E
0
|
2
p
2
πδ
p
ω
q`
̄
S
φφ
p
ω
qq
,
(23)
where we’ve also used the realness of
φ
p
t
q
to set
S
φφ
p
ω
q “
̄
S
φφ
p
ω
q
.
This relates the phase noise power spectral density to the optical power spectum
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
p
ω
q “
ˆ
a
in
p
ω
q`
a
in
p
ω
q
,
(24)
where
a
in
p
ω
q
is a stochastic input with
x
a
:
in
p
ω
q
a
in
p
ω
1
qy “
S
EE
p
ω
q
δ
p
ω
`
ω
1
q
(here the
averages used for correlation functions correspond to classical ensemble averages and
a
:
in
p
ω
q ” p
a
in
p ́
ω
qq
̊
).
There is, however, an additional subtlety when performing mechanical mode
thermometry with a laser beam affected by phase noise; correlations between the positive
Laser noise in cavity-optomechanical cooling and thermometry
11
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
p
t
q
»
E
0
`
e
c
cos
ωt
`
s
sin
ωt
̆
»
E
0
ˆ
1
`
1
2
p
β
s
`
c
q
e
iωt
́
1
2
p
β
s
́
c
q
e
́
iωt
̇
.
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
p ́q
in
p
ω
q “ ́
́
a
p`q
in
p ́
ω
q
̄
̊
(25)
where
a
p ́q
in
p
ω
q “
θ
p ́
ω
q
a
in
p
ω
q
, and
a
p`q
in
p
ω
q “
θ
p
ω
q
a
in
p
ω
q
. The total phase noise signal
can then expressed as
a
in
p
ω
q “
a
p`q
in
p
ω
q ́
a
p`q:
in
p ́
ω
q
. 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
x
a
p`q:
in
p
ω
q
a
p`q
in
p
ω
1
qy “
S
EE
p
ω
q
δ
p
ω
`
ω
1
q
θ
p
ω
q
,
(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 eqn. (5) for
ˆ
b
p
ω
q
,
replacing ˆ
a
in
p
ω
q
with ˆ
a
in
,
tot
p
ω
q
which includes the classical phase noise on the input laser.
From the non-zero correlation
x
ˆ
a
:
in
,
tot
p
ω
q
ˆ
a
in
,
tot
p
ω
1
qy “
S
EE
p
ω
q
δ
p
ω
`
ω
1
q
for
ω,ω
1
ą
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
EE
p
ω
q
in the noise spectrum of the mechanical motion given by eqn. (8),
S
bb
p
ω
q “
γn
f,φ
p
ω
q
p
ω
m
`
ω
q
2
`p
γ
{
2
q
2
,
(27)
where
n
f,φ
p
ω
q
|
ω
m
γ
i
n
b
γ
`
|
G
|
2
κ
γ
1
`p
κ
e
{
2
κ
q
S
EE
p
ω
q
p
́
ω
q
2
`p
κ
{
2
q
2
`
|
G
|
2
κ
γ
p
κ
e
{
2
κ
q
S
EE
p
ω
q
p
`
ω
q
2
`p
κ
{
2
q
2
.
(28)
As before, assuming a high mechanical
Q
-factor and the driven weak-coupling regime
p
κ
"
γ
q
, we substitute
n
f,φ
p ́
ω
m
q
for
n
f,φ
p
ω
q
and relate it to the average mode occupancy
in the presence of laser phase noise,
x
ˆ
n
y
φ
,
x
ˆ
n
y
φ
|
ω
m
γ
i
n
b
γ
`
γ
OM
γ
«
ˆ
κ
4
ω
m
̇
2
`
́
κ
e
2
κ
̄
n
φ
ff
,
(29)