Journal of Modern Optics
Vol. 55, Nos. 19–20, 10–20 November 2008, 3329–3338

Quantum theory of optomechanical cooling
Florian Marquardta*, A.A. Clerkb and S.M. GirvincaDownloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009

Department of Physics, Center for NanoScience and Arnold-Sommerfeld Center for
Theoretical Physics, Ludwig-Maximilians-Universität München, München, Germany;b
Department of Physics, McGill University, Montreal, QC, Canada; cDepartment of Physics,
Yale University, New Haven, CT, USA(Received 7 March 2008; final version received 3 September 2008)
We review the quantum theory of cooling of a mechanical oscillator subject to theradiation pressure force due to light circulating inside a driven optical cavity.
Such optomechanical setups have been used recently in a series of experiments by
various groups to cool mechanical oscillators (such as cantilevers) by factors
reaching 105, and they may soon go to the ground state of mechanical motion.
We emphasize the importance of the sideband-resolved regime for ground statecooling, where the cavity ring-down rate is smaller than the mechanicalfrequency. Moreover, we illustrate the strong coupling regime, where the coolingrate exceeds the cavity ring-down rate and where the driven cavity resonance andthe mechanical oscillation hybridize.
Keywords: cavity QED; optomechanics; micromechanics; sideband cooling;radiation pressure

1. Introduction
The interaction of light with matter has been at the heart of the development of quantum
mechanics since its inception. As for the mechanical effects of light, these become mostpronounced in a setup where the light intensity is resonantly enhanced (i.e. an opticalcavity) and where photons transfer maximum momentum to a mechanical object, e.g. by
being reflected multiple times from a movable mirror attached to a cantilever. The studyof radiation pressure effects on a movable mirror was pioneered in seminal papers
by Braginsky and co-workers [1,2]. Strong changes of the mechanical properties of themirror were observed later in an experiment by the Walther group [3], wherea macroscopic mirror was found to exhibit two stable equilibrium positions under theaction of the cavity’s radiation field. The most recent series of activities in this field started
with experiments observing optomechanical cooling first using feedback [4,5] and later[6–12] using the intrinsic effect discussed in the following, i.e. ‘passive’ cooling, without
active feedback, using purely the radiation pressure backaction. In addition, we note theself-induced optomechanical oscillations [13] that have been observed in radiation-pressure
driven microtoroidal optical resonators [14,15] and other setups [16,17]. For a recent
review see [18]. The study of these systems has been made even more fruitful by the*Corresponding author. Email: florian.marquardt@physik.lmu.de
ISSN 0950–0340 print/ISSN 1362–3044 online
ß 2008 Taylor & Francis
DOI: 10.1080/09500340802454971
http://www.informaworld.com

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F. Marquardt et al.

realization that the same (or essentially similar) physics may be observed in systems
ranging from driven LC circuits coupled to cantilevers [19] over superconducting single
electron transistors and microwave cavities coupled to nanobeams [20–26] to cloudsof cold atoms in an optical lattice, whose oscillations couple to the light field [27,28].
Cooling to the ground-state may open the door to various quantum effects in thesesystems, including ‘cat’ states [29], entanglement [30,31], quantum nonlinear dynamics [32],and Fock state detection [12]. Therefore, optomechanical ground-state cooling is currently
probably the highest priority in the field.
All the intrinsic optomechanical cooling experiments are based on the fact thatthe radiation field introduces extra damping for the cantilever. In such a classical picture,
the extra damping reduces the effective temperature of the single mechanical modeof interest, while leaving the bulk temperature of the cantilever the same (in the absenceof appreciable light absorption). The resulting effective temperature is related to the bathtemperature T by Teff/T ¼ GM/(Gopt þ GM), where GM and Gopt are the intrinsic mechanicaldamping rate and the optomechanical cooling rate, respectively. Thus, there is no limit tocooling in this regime, provided the laser power (and thus the cooling rate Gopt) can beincreased without any deleterious effects such as unwanted heating by absorption, andprovided the cooling rate remains sufficiently smaller than the mechanical frequency andthe cavity ring-down rate. However, at sufficiently low temperatures, the unavoidable
photon shot noise inside the cavity counteracts cooling. To study the resulting quantum
limits to cooling, a fully quantum-mechanical theory is called for, which we provided
in [33], based on the general quantum noise approach. Independently, a derivationemphasizing the analogy to ion sideband cooling was developed in [34]. In the present
paper, we will review and illustrate our theory. We start by outlining the basic classicalpicture, then present the quantum noise approach that provides a transparent andstraightforward way to derive cooling rates and quantum limits for the phonon number.
Finally, we illustrate the strong coupling regime that was first predicted in [33].

2. Basic classical picture
In this section, we briefly review the basic classical description of optomechanical cooling.
This was pioneered by Braginsky and co-workers in their work on optomechanicaldamping [1,2], while the associated cooling effect was first demonstrated only rather
recently [7].
The main ingredient for optomechanical cooling is the appearance of extra damping,ideally without extra fluctuations. This damping is introduced because the light-induced
force reacts with a finite delay time. In the case of radiation pressure, this comes about due
to the ring-down time 1 of the cavity. On the other hand, bolometric (i.e. photothermal)
forces are produced when a bimorph cantilever absorbs some of the radiation circulating
inside the cavity. When bolometric forces dominate, it is the finite time of thermal
conductance that sets the time-lag between the impinging radiation intensity and theresulting change in the cantilever temperature, which is proportional to the force.
The physical picture behind damping is simplest when the time-lag is small compared to theoscillation period of the cantilever. Then both radiation pressure and bolometric forces giverise to the same physics, modulo the appearance of a different time-scale in the two cases.

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Journal of Modern Optics

To describe this physics, we first fix our coordinate system: increasing the displacement x of thecantilever means elongating the cavity, and thus the optical resonance frequency (of the modeof interest) decreases. Let us consider the cantilever being placed at some location to the left of
the resonance (Figure 1). This means that the optical mode frequency is still higher than thefrequency of the incoming laser radiation, which is therefore red-detuned with respect to theoptical resonance. Now imagine moving the cantilever in a small cycle. As it moves towards theresonance with a finite speed, the light-induced force does work on the cantilever. However,
due to the time-lag it remains smaller than it would be in the case of infinitely slow (adiabatic)
motion. Conversely, as the cantilever moves back again in the second half of the cycle, the force
extracts energy and it is larger than for adiabatic motion. In total, the work done during sucha cycle by the light-induced force is negative, i.e. mechanical energy is extracted from thecantilever (Figure 1(b)). This kind of physics may be modeled by writing down a simple
relaxation-type equation for the force, which tries to reach its proper x-dependent value F (x)with some time-lag:_ ¼ F ðxðtÞÞ  FðtÞ :
FðtÞ


ð1Þ

The cantilever is a damped harmonic oscillator driven by the light-induced force:mx€ ¼ m!2M ðx  x0 Þ  mGM x_ þ F,

ð2Þwhere x0 is the mechanical equilibrium position in the absence of radiation pressure, m isthe cantilever mass, !M is its mechnical frequency, and GM is the intrinsic damping rate.
Linearizing Equation (1) with respect to small displacements from the mechanicalequilibrium position x (calculated in the presence of radiation pressure) and inserting it
into the equation of motion ofÐ the cantilever then yields the extra damping force.
In Fourier space, where x(t) ¼ x[!]exp(i!t)d!/2, we find the following linearized
equation of motion (at ! 6¼ 0):

!2 mx½! ¼ m!2M x½! þ im!GM x½! þ F 0 ðxÞx½!=ð1
 i!Þ:ð3Þ

Comparing the last two terms (the intrinsic damping with the imaginary part of the
optomechanical term), we find that the optomechanical damping rate is given by
Gopt ¼(a)radiationpressureforceF 0 ðxÞ!M ,
m!M 1 þ ð!M Þ2cantilever (b)

ð4Þ

(c)

laser
optical cavity

Figure 1. (a) The standard optomechanical setup treated in the text: a driven optical cavity witha movable mirror. (b) Moving the mirror in a cycle can result in work extracted by the light-field, due
to the finite cavity ring-down rate. (c) Asymmetric quantum noise spectrum for the radiationpressure force noise, according to Equation (14). (The color version of this figure is included in the
online version of the journal.)Downloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009

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F. Marquardt et al.

for the simple ansatz of Equation (1). According to this analysis, one would expect
the maximum effect to occur when !M ¼ 1, i.e. when the time-delay matches the period of
the cantilever motion. As we will see further below, this conclusion is not upheld by the fullquantum-mechanical analysis for the case of radiation pressure.
Equation (1) and the subsequent analysis holds exactly for the bolometric force. Inthat case,  is the finite time of thermal conductance and F (x) ¼ F maxI(x)/Imaxis the displacement-dependent bolometric force, where I(x) ¼ Imax/(1 þ (2D(x)/)2) is theintensity profile and D(x) ¼ x!R/L is the position-dependent detuning between incoming
laser radiation and optical resonance at x ¼ 0. For the radiation-pressure force, we can use
the present analysis only in the regime of   !M, and only if we allow for a position
dependent relaxation rate 1/ (see [33]).
In both cases, however, the shape of Gopt as a function of cantilever position x isdetermined by the slope of the intensity profile, i.e. in particular by the sign of F 0 .
To the left of the resonance, where F 0 4 0, we indeed obtain extra damping: Gopt 4 0.
As long as there are no extra fluctuations introduced by the light-induced force (i.e. if
we may disregard shot noise), the effective temperature of the mechanical degree of
freedom is therefore reduced according to the ratio of intrinsic and optomechanicaldamping rates:
Teff ¼ T

GM:GM þ Gopt

ð5Þ

This can be obtained, for example, by solving the Langevin equation that includes thethermal fluctuations of the mechanical heat bath (whose strength is set by GM
according to the fluctuation-dissipation theorem). Then the effective temperature may
 2 i ¼ kB Teff where !~ M
be defined according to the equipartition theorem: m!~ 2M hðx  xÞ
contains the frequency renormalization due to the real part of the optomechanical termin Equation (3).

3. Quantum noise approach
The quantum regime is reached once the simple classical relation for the effective
temperature, Equation (5), ceases to be valid. This happens when one has to take into
account the shot noise that tends to heat the cantilever motion. It enforces a lower boundfor the cantilever phonon number that can be reached by optomechanical cooling, i.e.
a finite quantum limit, which we are going to calculate. Crucially, we will also show thata proper choice of parameters can make this bound on the phonon number become muchsmaller than one, indicating that ground state cooling is possible.
The quantum picture can also be understood in terms of Raman scattering: incoming
photons, red-detuned with respect to the optical resonance, absorb a phonon from thecantilever, thereby cooling it. However, there is also a finite probability for phononemission, and thus heating. The purpose of a quantum theory is to discuss the balance of
these effects.
The idea behind the quantum noise approach to quantum-dissipative systems is todescribe the environment fully by the correlator of the fluctuating force that couples to thequantum system of interest. If the coupling is weak enough, knowledge of the correlator isDownloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009

Journal of Modern Optics3333

sufficient to fully describe the influence of the environment. Essentially, the results of thisapproach are based on leading-order perturbation theory, i.e. Fermi’s Golden Rule andrelated expressions. It yields rates of second order in the coupling that can be used inside
standard master equations [35], to calculate the steady-state occupations and otherquantities of interest. In our case, this means looking at the spectrum of the radiationpressure force fluctuations, which are produced by the shot noise of photons inside thedriven optical cavity mode, i.e. a nonequilibrium environment. In other applications,
we might be dealing with the electrical field fluctuations produced, e.g. by a driven
electronic circuit (superconducting single electron transistor, quantum point contact,
LC circuit) capacitively coupled to some nanobeam. The general formulas remain the samefor all of these cases, and only the noise spectrum changes.
The Fourier transform of the force correlator defines the spectral noise density:ðDE^ Fð0Þ
^:ð6Þ
SFF ð!Þ ¼ dt expði!tÞ FðtÞThe noise spectrum SFF is real-valued and non-negative. However, in contrast to the^ and Fð0Þ^classical case, it is asymmetric in frequency, since FðtÞdo not commute. Thisasymmetry has an important physical meaning: contributions at positive frequencies
indicate the possibility of the environment to absorb energy, while those at negative
frequencies imply its ability to release energy (to the cantilever). All the optomechanicaleffects can be described in terms of SFF, as long as the coupling is weak.
The optomechanical damping rate is given by the difference of noise spectra at positive
and negative frequencies,
Gopt ¼

x2ZPF½SFF ð!M Þ  SFF ð!M Þ:
h2


ð7Þ

h/(2m!M)]1/2 is the zero-point amplitude of mechanical motion. This formula
Here xZPF ¼ [is obtained by applying Fermi’s Golden Rule to derive the transition rates arising fromthe coupling of the cantilever to the light field, i.e. from the term H^ int ¼ F^x^ in the
Hamiltonian. These are
Gopt
# ¼

x2ZPF
SFF ð!M Þ ,
h2


Gopt
" ¼

x2ZPF
SFF ð!M Þ:
h2

ð8Þ

These rates enter the complete master equation for the density matrix ^ of the cantilever inthe presence of the equilibrium heat bath (that would lead to a thermal population n th ) andthe radiation field:
h
i
opt
y

^
^
^

þ
G
ð
n
þ
1ÞÞD½
a
þ
ðG
þ
G
ÞD½
a

:
ð9Þ
_^ ¼ ðGopt
n
M
th
M
th
#
"
Here the equation has been written in the interaction picture (disregarding the oscillations
at !~ M ), and^ ^ ¼ 1 ð2A^ ^ A^ y  A^ y A^ ^  ^ A^ y AÞ
^
D½A
2

ð10Þ

3334

F. Marquardt et al.

is the standard Lindblad operator for downward (Â ¼ â) or upward (Â ¼ ây) transitions in
the oscillator. Restricting ourselvesto the populations nn, we obtain the equation for theP^ ¼ n nnn :phonon number n ¼ hni
n_ ¼ GM n th þ Gopt"  ðGM þ Gopt Þn,

ð11Þ

which yields the steady-state phonon number in the presence of optomechanicalcooling:

Downloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009n M ¼

GM n th þ Gopt n O
M:
GM þ Gopt

ð12Þ

This is the weighted average of the thermal and the optomechanical phonon numbers.
It represents the correct generalization of the classical formula for the effective
temperature, Equation (5). Here
nO
M ¼


1
Gopt
1
SFF ð!M Þ"1¼¼ opt opt
SFF ð!M Þ
Gopt G# =G"  1

ð13Þ

is the minimal phonon number reachable by optomechanical cooling. This quantum limit
is reached when Gopt  GM. Then, the cooling effect due to extra damping is balanced by
the shot noise in the cavity, which leads to heating.
^ A
The radiation pressure force is proportional to the photon number: F^ ¼ ðh!R =LÞa^ y a.
brief calculation for the photon number correlator inside a driven cavity [33] yields its
spectrum in the form of a Lorentzian that is shifted by the detuning D ¼ !L  !R between
laser and optical resonance frequency !R:
SFF ð!Þ ¼


h!R 2,np
2Lð! þ DÞ þ ð=2Þ2

ð14Þ

where n p is the photon number circulating inside the cavity. This asymmetric quantum
noise spectrum is illustrated in Figure 1(c), while a plot of the resulting steady-statephonon number is shown in Figure 2(a).
Inserting this spectrum into Equations (7) and (13) yields the optomechanical coolingrate and the minimum phonon number as a function of detuning D. The minimum of nO
M isreached at a detuning D ¼ ½!2M þ ð=2Þ2 1=2 , and it is (see Figure 2(b)):
0"1
2 #1=2
1

@ 1þ
1A:
ð15Þ
min n O
M ¼
2
2!MFor slow cantilevers, !M  , we have min n O
M ¼ =ð4!M Þ  1. Ground-state cooling
becomes possible for high-frequency cantilevers (and/or high-finesse cavities), when  !M. Then, we find
min n O
M 




4!M

2
:

ð16Þ

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Journal of Modern Optics
cantilever
phononnumber(log.)

(b)

1.0

tin

0 g rad

ula

rc

Ci

0.1

iat
ion

2

Downloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009

w

0
10 er

po

1
0

Minimum phonon number

(a)

100

sidebandresolved
regime

10
1
0.1
0.01
0.001
0.0001
0.001

0.01

0.1

1

10

100

g

D

in
etun

Mechanical frequency vs. ring-down rate

Figure 2. (a) Cantilever phonon number as a function of circulating radiation power inside thecavity and as a function of detuning between laser and optical resonance. The mean phonon numberin steady state is plotted on a logarithmic scale. Contour lines indicate n ¼ 1 and n ¼ 0:1. For thisplot, the following parameters have been used: !M = ¼ 0:3, nth ¼ 103 , Q ¼ !M =GM ¼ 106 , and(!R/!M)(xZPF/L)  0.012. (b) The minimum phonon number as a function of the ratio between themechanical frequency !M and the optical cavity’s ring-down rate , according to Equation (15).
Ground-state cooling is possible in the regime !M  , i.e. the ‘good cavity’ or ‘resolved sideband’
limit. (The color version of this figure is included in the online version of the journal.)

As explained in [34], these two regimes can be brought directly into correspondence withthe known regimes for laser-cooling of harmonically bound atoms, namely the Doppler
limit for !M   and the resolved sideband regime for !M  . In the resolved sideband
limit the Stokes and anti-Stokes lines reflected from the cantilever–cavity setup, which are
at !L  !M, can be resolved from the main line at !L. Only in this limit is ground statecooling possible.

4. Strong coupling effects
Up to now, we have assumed that the coupling between light and mechanical degree of
freedom is sufficiently weak to allow for a solution in terms of a master equation,
employing the rates obtained from the quantum noise approach. However, as the coupling
becomes stronger (e.g. by increasing the laser input power), Gopt may reach the cavity
decay rate . Then, the spectrum of force fluctuations is itself modified by the presence ofthe cantilever. It becomes necessary to solve for the coupled dynamics of the light field andthe mechanical motion. This has been done in [33], by writing down the Heisenberg
equations of motion for the cantilever and the optical mode, and solving them after
linearization. Here we will only discuss the result.
To analyze these features, let us consider the spectrum of the cantilever motion,
ð

^
,
ð17Þ
Scc ð!Þ ¼ dt expði!tÞ c^y ðtÞcð0Þ
where c^ is the annihilation operator for the cantilever harmonic oscillator. At weakcoupling, this spectrum displays a peak at the (renormalized) cantilever frequency, i.e. at! ¼ !~ M (the minus sign is a consequence of our choice of the definition, following [33]).

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F. Marquardt et al.
1.5
cantilever spectrum

1000
100

cantilever

1.0

ca
n
iv
e
dr

Downloaded By: [Canadian Research Knowledge Network] At: 15:34 7 January 2009vi
tyFrequency

10
1

0.5
0.5

1.0

1.5
1.5

Detuning

Figure 3. The spectrum Scc(!) of the cantilever motion, as a function of detuning D and spectral
frequency !, for fixed circulating power. For the plot shown here, the strong-coupling regime isreached, i.e. Gopt/ becomes larger than one near resonance. Consequently, the driven cavity
resonance and the cantilever resonance hybridize, and an avoided crossing is observed.
The parameters for this plot are: /!M ¼ 0.1, GM/!M ¼ 105, (!R/!M)(xZPF/L) ¼ 0.01, np ¼ 100,and n th ¼ 103 , and the color scale indicates the values of !MScc(!). (The color version of this figure
is included in the online version of the journal.)Ð
Its width (FWHM) is given by Gopt þ GM, and its total weight Scc(!)d!/2 yields the^ . As the laser power is increased, the width increases and thephonon number n ¼ hc^y ci
weight diminishes (in the cooling regime).
When Gopt/ is no longer much smaller than one, deviations from the weak-coupling
results start to appear [33]. The most dramatic effect is observed when Gopt/ 4 1/2: thepeak splits into two (Figure 3). As it were, in this strong-coupling regime one actuallyencounters the hybridization of two coupled harmonic oscillators, namely the cantilever
and the driven optical mode (with an effective frequency set by the detuning).
At resonance, i.e. for D ¼ !M, the splitting is set by 2, where the coupling frequency
 is determined by the circulating laser power, the ratio of mechanical zero-point
fluctuations to the cavity length, and by the optical resonance frequency:
xZPF: ¼ !R n 1=2
ð18Þ
pL
5. Outlook
During the past year, several new ideas have been introduced into the fieldof optomechanical cooling. For example, placing a movable membrane in the middle ofa standard optical cavity [12] can lead to orders of magnitude better performance, as itJournal of Modern Optics

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separates the mechanical from the optical elements. Such a setup has been used to coolfrom 300 K down to 7 mK, and it may ultimately by employed for Fock state detectionof mechanical vibrations [12]. Even nanomechanical objects (such as nanowires) might beplaced inside the standing light wave [36] and cooled by scattering. Doppler cooling of
Bragg mirrors may provide another promising approach [37]. Recent experiments on
nanomechanical resonators coupled to on-chip microwave transmission line resonators[26,38] and on trapped cold atoms interacting with a light field [28] point towards
promising new avenues in the field of optomechanics (see ‘News and Views’ [39]).
Furthermore, sideband-resolved cooling with !M/  20 has been demonstrated recently[40], paving the way for ground-state cooling when combined with cryogenics [41].

Acknowledgements
We thank Jack Harris for sharing his many insights about optomechanical systems. F.M. also
thanks B. Kubala, M. Ludwig and C. Neuenhahn for discussions on this topic. F.M. acknowledges
the EuroSQIP program and the DFG for support via SFB 631, the Emmy-Noether program, and the
NIM cluster of excellence. The work of S.M.G. was supported by the NSF via grants DMR-0603369
and DMR-0653377. A.A.C. would like to thank the Canadian Institute for Advanced Research(CIFAR).

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