Cavity optomechanics in new regimes and with new devices · Cavity optomechanics in new regimes and with new devices 1 ... one moveable mirror captures the main idea of such setups,

Cavity optomechanics in new regimes and with new devices

1. What? Why? How?

2. Single-photon regime

3. Dissipative coupling

Andreas NunnenkampUniversity of Basel

P. Meystre: A short walk through quantum optomechanics

structure rather than via the internal structure of materi-

als. This could be for example an optical resonator with

a series of narrow resonances, or an electromagnetic res-

onator such as a superconducting LC circuit. Indeed, nu-

merous designs can achieve optomechanical control via

radiation pressure effects in high-quality resonators. They

range from nanometer-sized devices of as little as 107

atoms to micromechanical structures of 1014

atoms and

to macroscopic centimeter-sized mirrors used in gravita-

tional wave detectors.

That development first appeared at the horizon in the

1960s, but more so in the late 1970s and 1980s. It was

initially largely driven by the developments in optical

gravitational wave antennas spearheaded by V. Bragin-

sky, K. Thorne, C. Caves, and others [2, 3, 8]. These anten-

nas operate by coupling kilogram-size test masses to the

end-mirrors of a large path length optical interferometer.

Changes in the optical path length due to local changes

in the curvature of space-time modulate the frequency of

the cavity resonances and in turn, modulate the optical

transmission through the interferometer. It is in this con-

text that researchers understood fundamental quantum

optical effects on mechanics and mechanical detection

such as the standard quantum limit, and how the basic

light-matter interaction can generate non-classical states

of light.

Braginsky and colleagues demonstrated cavity op-

tomechanical effects with microwaves [9] as early as 1967.

In the optical regime, the first demonstration of these

effects was the radiation-pressure induced optical bista-

bility in the transmission of a Fabry-Pérot interferometer,

realized by Dorsel el al. in 1983 [10]. In addition to these

adiabatic effects, cooling or heating of the mechanical mo-

tion is also possible, due to the finite time delay between

the mechanical motion and the response of the intracavity

field, see section 2.2. The cooling effect was first observed

in the microwave domain by Blair et al. [11] in a Niobium

high-Q resonant mass gravitational radiation antenna,

and 10 years later in the optical domain in several labo-

ratories around the world: first via feedback cooling of

a mechanical mirror by Cohadon et al. [12], followed by

photothermal cooling by Karrai and coworkers [13], and

shortly thereafter by radiation pressure cooling in several

groups [14–19]. Also worth mentioning is that as early as

1998 Ritsch and coworkers proposed a related scheme to

cool atoms inside a cavity [20].

This paper reviews the basic physics of quantum op-

tomechanics and gives a brief overview of some of its

recent developments and current areas of focus. Section

2 outlines the basic theory of cavity optomechanical cool-

ing and sketches a brief status report of the experimental

state-of-the-art in ground state cooling of mechanical os-

Figure 1 (Color online) Generic cavity optomechanical system.The cavity consists of a highly reflective fixed input mirror anda small movable end mirror harmonically coupled to a supportthat acts as a thermal reservoir.

cillators, a snapshot of a situation likely to be rapidly out-

dated. Of course ground state cooling is only the first step

in quantum optomechanics. Quantum state preparation,

control and characterization are the next challenges of

the field. Section 3 gives an overview of some of the major

trends in this area, and discusses topics of much current

interest such as the so-called strong-coupling regime, me-

chanical squeezing, and pulsed optomechanics. Section 4

discusses a complementary “bottom-up” approach that

exploits ultracold atomic samples instead of nanoscale

systems to study quantum optomechanical effects. Finally,

Section 5 is an outlook that concentrates largely on the

functionalization of quantum optomechanical systems

and their promise in metrology applications.

2 Basic theory

To describe the basic physics underlying the main aspects

of cavity optomechanics it is sufficient to consider an op-

tically driven Fabry-Pérot resonator with one end mirror

fixed -and effectively assumed to be infinitely massive,

and the other harmonically bound and allowed to oscil-

late under the action of radiation pressure from the in-

tracavity light field of frequency ωL , see Fig. 1. Braginsky

recognized as early as 1967 [9] that as radiation pressure

drives the mirror, it changes the cavity length, and hence

the intracavity light field intensity and phase. This results

in two main effects: the ”optical spring effect,” an opti-

cally induced change in the oscillation frequency of the

mirror that can produce a significant stiffening of its effec-

tive frequency; and optical damping, or “cold damping,”

whereby the optical field acts effectively as a viscous fluid

that can damp the mirror motion and cool its center-of-

mass motion.

2 Copyright line will be provided by the publisher

Introduction

Cavity optomechanics

radiation pressure force

cavity mode mechanical oscillator

Cavity optomechanics: the mechanical oscillator

H = ωM

b†b+

1

2

x = xZPF(b+ b†)

phonons with

thermal state

nth =kBT

ωM

Cavity optomechanics: the cavity mode

photonsL

Cavity optomechanics: the optomechanical coupling

L

“Standard model of optomechanics”

Marquardt & Girvin, Physics 2, 40 (2009)Kippenberg & Vahala, Science 321, 1172 (2008)

H = ωC(x)a†a+ ωM b

†b


N.B. We neglect the dynamical Casimir effect.

radiation-pressure force

with

withx = xZPF(b+ b†)


Introduction

The field of optomechanics deals with systems in which optical and mechanical degreesof freedom are coupled to each other. Interaction between light and mechanical ele-ments was already considered centuries ago by Johannes Kepler, who studied the tailsof comets. Nowadays it is well known that light exerts a radiation pressure force onobjects. The momentum of the impinging light is transferred to the mechanical objectand thus influences its motion. Although the action of sunlight on comets looks spectac-ular and impressive, the force due to a single photon is small compared to the inertia ofmacroscopic objects. Nevertheless, radiation pressure has to be taken into account forprecision measurements and is a major issue e.g. for the detection of gravitational waves.To make use of the radiation pressure force, it is convenient to use cavities to enhancethe light intensity and thereby the coupling strength. A typical optomechanical setupconsists of a cavity with one fixed and one moveable end-mirror, see Fig. 0.1 (a). Thelight circulating in the cavity displaces the moveable mirror and thus changes the cavitylength. This, in turn, shifts the resonance frequency of the cavity, which influences theintensity and thereby the force on the mirror. In general, systems where the coupling be-tween optical and mechanical degree of freedom arises due to a displacement-dependentcavity frequency are called dispersively coupled. The generic picture of a cavity withone moveable mirror captures the main idea of such setups, however many di!erent ex-perimental realizations exist, including e.g. cantilevers, membranes or microcavaties (seeFig. 0.2) with frequencies from Hz to GHz and masses from 10!20g to kg [1].

!"#

!"#$%& '(%)*+$%&

!"#$%*,-.*#) '(%)+$%*,-.*#)

!"#$%!"#!'(%)*+$%*,$+#(/*%#$!+

0.1

Figure 0.1.: (a) Typical optomechanical system implementing dispersive coupling be-tween optics and mechanics. (b) Schematic picture illustrating the generalsetup envisioned for dispersively coupled systems, also including coupling tothe environment.

3

∆/ωm

ωm/κ ωm/γ

g/κ g/ωm

Ω/κwith the dimensionless parameters

good/bad-cavity limit mechanical quality factor

cavity shift per phonon in units of line width

oscillator displacement per photon in units of its ZPF

drive strength & detuning

Single-photon regime

The single-photon strong-coupling regime

D. Stamper-Kurn (Berkeley)

g/ωM ∼ 2 ωM/κ ∼ 0.1Brooks et al., Nature 488, 476 (2012)

5

b

2 µm

FIG. 1. Schematic description of the experiment. a, Colorized optical micrograph of the microwave resonator formed by a spiralinductor shunted by a parallel-plate capacitor. b, Colorized scanning electron micrograph shows the upper plate of the capacitor issuspended ∼50 nm above the lower plate and is free to vibrate like a taught, circular drum. The metallization is sputtered aluminum(blue) patterned on a sapphire substrate (black). c, This circuit is measured by applying microwave signals near the electrical resonancefrequency through resistive coaxial lines. The outgoing signals, in which the mechanical motion is encoded as modulation sidebands ofthe applied microwave tone, is coupled to a low noise, cryogenic amplifier via a superconducting coaxial cable. Cryogenic attenuatorson the input line and isolators on the output line ensure that thermal noise is reduced below the vacuum noise at microwave frequencies.Teufel et al., Nature 475, 359 (2011)

g/ωM ∼ 5× 10−6 ωM/κ ∼ 50J. Teufel / K. Lehnert (Boulder)

O. Painter (Caltech)

ωM/κ ∼ 8g/ωM ∼ 2.5× 10−4

Chan et al., Nature 478, 89 (2011)

ωM/κ ∼ 30g/κ ∼ 0.007− 0.1Chan et al., arXiv:1206.2099

What happens when and ?g/ωM ∼ 1 ωM/κ ∼ 1

with


Completing the square

new equilibrium position of mechanical oscillator

photon-photon interaction

Interaction is diagonal, but drive and damping are not!

polaron transformation


Mahan, Many-Particle Physics (Springer)Nunnenkamp et al., PRL 107, 063602 (2011)

oscillator displacement per photon in units of its ZPF

The nonlinear spectrum leads to many interesting effects!

The energy spectrum is

2

−4 −2 0 2 4

0.2

0.4

0.6

0.8

1

!/!M

!a†a"

/n0

(a)

−6 −4 −2 0 20

0.5

1

1.5

2

!/!M

S(!

)/n

0

(b)

FIG. 2: (Color online) Detecting the single-photon strong-couplingregime: (a) Steady-state mean photon number !a†a" as a function ofdetuning ! and (b) power spectrum S(!) at ! = 0. !M = g and!M/" = 20 for all curves, !M/# = 2 and nth = 0 (blue solid),!M/# = 2 and nth = 1 (red dashed) as well as !M/# = 0.5 andnth = 0 (black dash-dotted). The thin black solid line in (a) showsthe empty cavity response g = 0 for comparison. n0 is the meanphoton number on resonance, i.e. n0 = 4"2/#2.

Note first that the Hamiltonian (1) conserves photon num-ber, i.e. [a†a, H0] = 0. The Hamiltonian in the subspace of nphotons is a harmonic oscillator with frequency !M which isdisplaced by!nx0/xZPF = !ng/!M . Thus, the eigenvaluesof (1) areEnm = !Rn!g2n2/!M+!Mmwith non-negativeintegers n and m. The anharmonicity is due to the optome-chanical interaction which is the product of photon numberand oscillator displacement gn " !nx0/xZPF. We show thespectrum and eigenfunctions of Hamiltonian (1) in Fig. 1 (b).In order to include drive and decay we use standard input-

output theory [25]. In a frame rotating at the frequency of theoptical drive, the non-linear quantum Langevin equations read

˙a = +i!a!"

2a! ig

!

b† + b"

a+#" ain (2)

˙b = !i!M b!#

2b! iga†a+

## bin. (3)

where ! = !L ! !R is the detuning between laser !L andresonator frequency !R, and # and " are the mechanical andcavity damping rates. The cavity input ain is a sum of a co-herent amplitude ain and a vacuum noise operator $ satisfy-ing $$(t)$†(t!)% = %(t ! t!) and $$†(t)$(t!)% = 0. Finally,we assume that the mechanical bath is Markovian and has atemperature T , i.e. $bin(t)b†in(t!)% = (nth + 1)%(t ! t!) and$b†in(t)bin(t!)% = nth%(t! t!) with nth & kBT/!!M .The model is characterized by three dimensionless param-

eters: the mechanical quality factor !M/#, the resolved-sideband parameter !M/", and the granularity parameter g/"[24, 26]. The latter is the cavity frequency shift in units of itslinewidth when the oscillator is displaced by one zero-pointuncertainty xZPF. Finally, g/!M = (g/") " (!M/")"1 isthe oscillator displacement in units of xZPF caused by the ra-diation pressure of a single photon. If g/!M ' 1 we will saythe system is in the single-photon strong-coupling regime.Approximate solution for weak drive. It is well known that

the HamiltonianH0 can be diagonalized by the polaron trans-form given by U = e"S with S = g

!Ma†a(b† ! b) [27]. Here

we use it to find an approximate solution to Eqs. (2) and (3).

In steady-state and for weak optical drive we obtain

a(t) =#"

# t

"#

d& e"("/2"i!)(t"#)eX(t)e"X(#)ain(&) (4)

where we defined ! = !+ g2/!M and X(t) is given by

X(t) =

##g

!M

# t

"#

d& e"$(t"#)/2$

e"i!M(t"#)bin(&) !H.c.%

(5)Using this analytic approach, we calculate properties of the

optical field. We get for the steady-state mean photon number

$a†a%n0

=#&

n=0

(g/!M )2n

4n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""("+ n#)e"(g/!M)2(2nth+1)

("+n$2 )2 + (!! (n! 2k)!M )2

(6)

and the cavity amplitude relevant for homodyne experiments

$a%#n0

=#&

n=0

(g/!M)2n

2n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""e"(g/!M)2(2nth+1)

("+n$2 )! i(!! (n! 2k)!M )

(7)

where n0 = 4"2/"2 with " =#"|ain| is the mean pho-

ton number for g = 0 on resonance ! = 0. The quanti-ties (6) and (7) are sums of resonances which are spaced bythe mechanical frequency !M . Let us discuss first the caseof zero temperature, i.e. nth = 0, when only terms withk = 0 contribute in Eqs. (6) and (7). In this case, the reso-nances are weighted by a Poisson distribution with variance(g/!M )2 and the widths are " + n#. The resonances can beunderstood as transitions between the vacuum state |0, 0% andthe manifold of one-photon eigenstates |1,m% of the Hamilto-nian (1). They are resonant if the laser frequency !L matches!L = E1m!E00 = !R!g2/!M +m!M . The Poission dis-tribution is due to the Franck-Condon factors |$m|eX |0%|2 =

|)

dx'$m(x! x0)'(x)|2 = (g/!M )2me"(g/!M )2/m! where

|m% is the state withm phonons,'m(x) is its real-space wave-function, and x0/xZPF = g/!M . At finite temperature thestates |0,m% with m > 0 are thermally occupied leading to aredistribution of weight among the peaks and additional reso-nances at !L = !R ! g2/!M !m!M .In the limit " ( # we obtain the cavity spectrum S(!) =

)#

"#dt [ei!t$a†(t)a(0)% ! |$a(t)%|2] as

S(!) =#&

m,n=0

Cmn(m+ n)#!

(m+n)$2

"2+ (! ! (m! n)!M )2

(8)

with the n = m = 0 term excluded. The coefficientsCmn areindependent of ! but rather involved and will not be shown.The optical output spectrum has sideband peaks at integer

multiples of the mechanical frequency !M whose widths are

E

g2

ωM

ωR

2ωR4g2

ωM

0 ωM

2g

ωM

2g

ωM

x/xZPF

a†a, H

= 0Photon number conserved

At fixed photon number n, the mechanicaloscillator is displaced by

−nx0/xZPF = −2ng/ωM

4 2 0 2 4

0.2

0.4

0.6

0.8

1

!/!M

!a† a"/

n0

E

g2

!M

!R

2!R4g2

!M

0 !M

!"#

2g

!M

2g

!M

x/xZPF

2

−4 −2 0 2 4

0.2

0.4

0.6

0.8

1

!/!M

!a†a"

/n

0

(a)

−6 −4 −2 0 20

0.5

1

1.5

2

!/!M

S(!

)/n

0

(b)




˙a = +i!a!"

2a! ig

!

b† + b"

a+#" ain (2)

˙b = !i!M b!#

2b! iga†a+

## bin. (3)




!Ma†a(b† ! b) [27]. Here



a(t) =#"

# t

"#

d& e"("/2"i!)(t"#)eX(t)e"X(#)ain(&) (4)


X(t) =

##g

!M

# t

"#

d& e"$(t"#)/2$

e"i!M(t"#)bin(&) !H.c.%



$a†a%n0

=#&

n=0

(g/!M )2n

4n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""("+ n#)e"(g/!M)2(2nth+1)

("+n$2 )2 + (!! (n! 2k)!M )2

(6)


$a%#n0

=#&

n=0

(g/!M)2n

2n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""e"(g/!M)2(2nth+1)

("+n$2 )! i(!! (n! 2k)!M )

(7)



|)



)#

"#dt [ei!t$a†(t)a(0)% ! |$a(t)%|2] as

S(!) =#&

m,n=0

Cmn(m+ n)#!

(m+n)$2

"2+ (! ! (m! n)!M )2

(8)



Single-photon regime: the cavity response

Nunnenkamp et al., PRL 107, 063602 (2011)also: Rabl, PRL 107, 063601 (2011) → solution of quantum Langevin equations

In the single-photon regime multiple resonances appear.

→ Franck-Condon factors

g = ωM

ωM/κ = 2,ωM/γ = 20, n0 = 4Ω2/κ2

Single-photon regime: the cavity output spectrum

Nunnenkamp et al., PRL 107, 063602 (2011)also: Liao et al., arXiv:1201.1696→ solution of quantum Langevin equations

ωM/κ = 2,ωM/γ = 20, n0 = 4Ω2/κ2

6 4 2 0 20

0.5

1

1.5

2

!/!M

S(!

)/n

0

g = ωM

nth = 0

In the single-photon regime multiple sidebands appear.

→ nonlinear photon-phonon coupling

S(ω) =

∞

−∞dt eiωta†(t)a(0)

E

g2

!M

!R

2!R4g2

!M

0 !M

!"#

2g

!M

2g

!M

x/xZPF

2

−4 −2 0 2 4

0.2

0.4

0.6

0.8

1

!/!M

!a†a"

/n

0

(a)

−6 −4 −2 0 20

0.5

1

1.5

2

!/!M

S(!

)/n

0

(b)




˙a = +i!a!"

2a! ig

!

b† + b"

a+#" ain (2)

˙b = !i!M b!#

2b! iga†a+

## bin. (3)




!Ma†a(b† ! b) [27]. Here



a(t) =#"

# t

"#

d& e"("/2"i!)(t"#)eX(t)e"X(#)ain(&) (4)


X(t) =

##g

!M

# t

"#

d& e"$(t"#)/2$

e"i!M(t"#)bin(&) !H.c.%



$a†a%n0

=#&

n=0

(g/!M )2n

4n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""("+ n#)e"(g/!M)2(2nth+1)

("+n$2 )2 + (!! (n! 2k)!M )2

(6)


$a%#n0

=#&

n=0

(g/!M)2n

2n!

n&

k=0

'

n

k

(

(nth + 1)n"knkth

""e"(g/!M)2(2nth+1)

("+n$2 )! i(!! (n! 2k)!M )

(7)



|)



)#

"#dt [ei!t$a†(t)a(0)% ! |$a(t)%|2] as

S(!) =#&

m,n=0

Cmn(m+ n)#!

(m+n)$2

"2+ (! ! (m! n)!M )2

(8)



Dissipative coupling

Dispersive vs. dissipative optomechanics

Elste et al., PRL 102, 207209 (2009)Xuereb et al., PRL 107, 213604 (2011)

Dissipative coupling

κ(x) = κ+∂κ

∂xx

cavity linewidth. The idea is as follows: Consider the MSIoperated at a point where the transmissivity ! of theeffective mirror is close to zero; ! will then depend sensi-tively on the membrane displacement x. Combining thiscompound ‘‘MSI mirror’’ with a perfect mirror in its darkport will result in an effective FPI whose linewidth dependson x dominantly via ! and not via the change in the(effective) cavity length L; see Fig. 1. In contrast to atrue FPI, therefore, g" and g! have a different functionaldependence on x. This feature gives rise to a topology(i) where the optomechanical interaction can be tuned atwill between being strongly dissipative or dispersive and(ii) which can realize dissipative optomechanics leading toground-state cooling of the mechanical oscillator via quan-tum noise interference, as discussed in Ref. [24]. Thedistinctive signature of this interaction, an asymmetricFano line shape, is observable in the spectrum of the cavityoutput field. We also report on the suitability of using thissystem for (iii) sensitive position transduction.

This Letter is structured as follows. We shall first de-scribe the physical model and write down the resultingHamiltonian and input-output relation. These are then usedto derive the equations of motion for the cavity field andoscillator motion. The resulting dynamics is solved toobtain the steady-state mechanical occupation number an-alytically in the weak-coupling limit and also numericallyin both weak- and strong-coupling regimes.

Model.—The Hamiltonian including both dispersive anddissipative effects in an opto- or electromechanical [24]system can be written as [34]

HOM ! "#!$ g!x%aya$!mbyb$

Zd!!ay!a!

$ i! """"""""

2"c

p$ g"""""""""

2"c

p x#Z d!"""""""

2#p "ay!a# H:c:%: (1)

This Hamiltonian is written in a frame rotating at theoptical frequency of some driving field !d; ! !!d #!c is the detuning from cavity resonance (!c) whoseannihilation operator is a. x ! "b$ by%=

"""2

pis the dimen-

sionless displacement, and b is the annihilation operator

for the mechanical oscillator of frequency !m. The secondline in Eq. (1) provides the coupling of the cavity mode a tothe modes a! of the external field coupling into and out ofthe effective FPI and will give rise to the finite width "c ofthe resonance. The two terms proportional to x describe theshifts of the cavity resonance and width with mechanicaldisplacement, with strengths characterized by g!;", respec-tively. This Hamiltonian generalizes the dispersive-onlyHamiltonian that is considered in most works on optome-chanical systems. Before we proceed to make the connec-tion of Hamiltonian (1) and the physics of a MSI [32,33],we note that the standard cavity-input–output relation mustbe generalized to accommodate the effect of dissipativecoupling [34]:

a out # ain ! &""""""""2"c

p$ "g"=

""""""""2"c

p%x'a: (2)

This relation provides a boundary condition connectinglight leaving the effective FPI with light entering it, andthe intracavity dynamics. It can be identified with

the familiar relation aout # ain !""""""""""""2""x%

p, taking ""x% !

"c $ g"x and truncating the square root to first order in x.The generalized optomechanical Hamiltonian (1) is real-

ized in the MSI shown in Fig. 1 operating close to a dark-port condition, i.e., when most light is directly reflected atthe first beam splitter (BS). The whole system can then bedescribed as an effective Fabry-Perot interferometer oflength L, operating in the good-cavity limit, formed be-tween the perfect end mirror M1 and an effective endmirror M; 2L is the length of the Sagnac modeM1-BS-M2-M3-BS. The reflectivity $ and transmissivity! of M depend on the complex reflectivity R (r) andtransmissivity T (t) of the BS (of the micromirror), aswell as the displacement % of M from the midpoint ofM2-M3:

$ ! #"R2re2i% $ T2re#2i% $ 2RTt%e#i argt; and (3a)

! ! &"RT(e2i% # c:c:%r# "jRj2 # jTj2%t'e#i argt: (3b)

Assuming the close-to-dark-port condition j!j ) 1, aquantization along the standard routes of cavity QED[34] yields a Hamiltonian of the form in Eq. (1) with theusual result "c ! #c="2L% lnj$j * cj!j2="4L%, as well asdispersive and dissipative couplings

g! ! #2"!cx0=L%&"jRj2 # jTj2% $ ! cos"argt%'; (4a)

g" ! #"""2

pij!jei&"!cx0=L%&2RT $ $ cos"argt%'; (4b)

where & * 0 close to resonance. These results guaranteethat the values of g! and g" can be controlled indepen-dently by choosing % (i.e., positioning the membrane) andthe reflectivity of the central BS appropriately. The needfor a sharp resonance demands jRj * jTj. We note, how-ever, that jRj ! jTj is required to be able to set g! ! 0with j!j> 0. In the following, we will specialize to thismost interesting case of a purely dissipative coupling.We will provide experimental case studies below and

FIG. 1 (color online). (a) Topology of the Michelson-Sagnacinterferometer. (b) Effective cavity; the properties of M dependon M.

PRL 107, 213604 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

18 NOVEMBER 2011

213604-2

L

C

Cout(x)

Dispersive coupling

L

C(x)

Cout

Sideband cooling: the quantum noise approach

Incoming photons, red-detuned with respect to the optical resonance, absorb a

phonon from the cantilever, thereby cooling it. However, there is also a finite

probability for phonon emission, 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 to describe the environment fully by the correlator of the fluctuating force

that couples to the quantum system of interest. If the coupling is weak enough,

knowledge of the correlator is sufficient to fully describe the influence of the

environment. In our case, this means looking at the spectrum of the radiation

pressure force fluctuations, which are produced by the shot noise of photons

inside the driven 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 same for all of these cases, and

only the noise spectrum changes.

The Fourier transform of the force correlator defines the spectral noise den-

sity:

SFF (ω) = dt eiωt

F (t)F (0)

. (6)

The noise spectrum SFF is real-valued and non-negative. However, in contrast

to the classical case, it is asymmetric in frequency, since F (t) and F (0) do not

commute. This asymmetry 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 optomechanical effects 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,

Γopt =x

2ZPF

2[SFF (ωM )− SFF (−ωM )] . (7)

This formula is obtained by applying Fermi’s Golden Rule to derive the transi-

tion rates arising from the coupling of the cantilever to the light field, i.e. from

the term Hint = −F x in the Hamiltonian. These are

Γopt↓ =

x2ZPF

2SFF (ωM ) , Γopt

↑ =x

2ZPF

2SFF (−ωM ). (8)

These rates enter the complete master equation for the density matrix ρ of the

cantilever in the presence of the equilibrium heat bath (that would lead to a

thermal population nth) and the radiation field:

˙ρ =

(Γopt

↓ + ΓM (nth + 1))D[a] + (Γopt↑ + ΓM nth)D[a

†]

ρ (9)

5


















sity:

SFF (ω) = dt eiωt

F (t)F (0)

. (6)










Γopt =x

2ZPF

2[SFF (ωM )− SFF (−ωM )] . (7)




Γopt↓ =

x2ZPF

2SFF (ωM ) , Γopt

↑ =x

2ZPF

2SFF (−ωM ). (8)




˙ρ =

(Γopt


†]

ρ (9)

5

At weak coupling all you need to know is the force spectrum

cantilever phonon number

(log.)

circulating radiation power detuning

0

12

1.0

0.1

100

0

(a)

min

imum

phonon n

um

ber

mechanical frequency vs. ring-down rate

0.001 0.01 0.1 1 10 100

0.0001

0.001

0.01

0.1

1

10

100

sideband-

resolved

regime

(b)

Figure 2: (a) Cantilever phonon number as a function of circulating radiation

power inside the cavity and as a function of detuning between laser and optical

resonance. The mean phonon number in steady state is plotted on a logarithmic

scale. Contour lines indicate n = 1 and n = 0.1. For this plot, the following

parameters have been used: ωM/κ = 0.3, nth = 103, Q = ωM/ΓM = 10

6, and

(ωR/ωM )(xZPF/L) ≈ 0.012. (b) The minimum phonon number as a function of

the ratio between the mechanical frequency ωM and the optical cavity’s ring-

down rate κ, according to Eq. (15). Ground-state cooling is possible in the

regime ωM κ, i.e. the “good cavity” or “resolved sideband” limit.

Here the equation has been written in the interaction picture (disregarding the

oscillations at ωM ), and

D[A]ρ =1

2(2AρA† − A†Aρ− ρA†A) (10)

is the standard Lindblad operator for downward (A = a) or upward (A = a†)transitions in the oscillator. Restricting ourselves to the populations ρnn, we

obtain the equation for the phonon number n = n =

nnρnn:

˙n = ΓM nth + Γopt↑ − (ΓM + Γopt)n, (11)

which yields the steady-state phonon number in the presence of optomechanical

cooling:

nM =ΓM nth + ΓoptnO

M

ΓM + Γopt. (12)

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, Eq. (5). Here

nO

M =Γopt↑

Γopt=

1

Γopt↓ /Γopt

↑ − 1=

SFF (ωM )

SFF (−ωM )− 1

−1

(13)

6


(log.)


0

12

1.0

0.1

100

0

(a)

min

imu

m p

ho

no

n n

um

be

r


0.001 0.01 0.1 1 10 100

0.0001

0.001

0.01

0.1

1

10

100

sideband-

resolved

regime

(b)






6, and







D[A]ρ =1

2(2AρA† − A†Aρ− ρA†A) (10)



nnρnn:



cooling:


M

ΓM + Γopt. (12)




nO

M =Γopt↑

Γopt=

1

Γopt↓ /Γopt

↑ − 1=

SFF (ωM )

SFF (−ωM )− 1

−1

(13)

6

Γopt = Γopt↓ − Γopt

↑

And calculate the steady-state phonon number

optical damping

minimal phonon number


















sity:

SFF (ω) = dt eiωt

F (t)F (0)

. (6)










Γopt =x

2ZPF

2[SFF (ωM )− SFF (−ωM )] . (7)




Γopt↓ =

x2ZPF

2SFF (ωM ) , Γopt

↑ =x

2ZPF

2SFF (−ωM ). (8)




˙ρ =

(Γopt


†]

ρ (9)

5


















sity:

SFF (ω) = dt eiωt

F (t)F (0)

. (6)










Γopt =x

2ZPF

2[SFF (ωM )− SFF (−ωM )] . (7)




Γopt↓ =

x2ZPF

2SFF (ωM ) , Γopt

↑ =x

2ZPF

2SFF (−ωM ). (8)




˙ρ =

(Γopt


†]

ρ (9)

5

Obtain the rates with Fermi’s Golden Rule

Clerk et al., RMP 82, 1155 (2010)Marquardt et al., PRL 99, 093902 (2007)

Wilson-Rae et al., PRL 99, 093901 (2007)

Sideband cooling: the quantum noise approach

Marquardt et al., PRL 99, 093902 (2007)Wilson-Rae et al., PRL 99, 093901 (2007)

Ground-state cooling is possible in the sideband-resolved regime ωM κ

Red-sideband cooling at

ωL = ωR − ωM

ωout = ωin + ωM


(log.)


0

12

1.0

0.1

100

0

(a)

min

imum

phonon n

um

ber


0.001 0.01 0.1 1 10 100

0.0001

0.001

0.01

0.1

1

10

100

sideband-

resolved

regime

(b)






6, and







D[A]ρ =1

2(2AρA† − A†Aρ− ρA†A) (10)



nnρnn:



cooling:


M

ΓM + Γopt. (12)




nO

M =Γopt↑

Γopt=

1

Γopt↓ /Γopt

↑ − 1=

SFF (ωM )

SFF (−ωM )− 1

−1

(13)

6

figure by F. Marquardt

is the minimal phonon number reachable by optomechanical cooling. This quan-

tum limit is reached when Γopt ΓM . Then, the cooling effect due to extra

damping is balanced by the shot noise in the cavity, which leads to heating.

The radiation pressure force is proportional to the photon number: F =

(ωR/L)a†a. A brief calculation for the photon number correlator inside a

driven cavity [32] yields its spectrum in the form of a Lorentzian that is shifted

by the detuning ∆ = ωL − ωR between laser and optical resonance frequency

ωR:

SFF (ω) =

ωR

L

2

npκ

(ω + ∆)2 + (κ/2)2, (14)

where np is the photon number circulating inside the cavity. A plot of the

resulting steady-state phonon number is shown in Fig. 2a.

Inserting this spectrum into Eqs. (7) and (13) yields the optomechanical

cooling rate and the minimum phonon number as a function of detuning ∆.

The minimum of nO

Mis reached at a detuning ∆ = −

ω2

M+ (κ/2)2, and it is

(see Fig. 2b):

min nO

M =1

2

1 +

κ

2ωM

2

− 1

. (15)

For slow cantilevers, ωM κ, we have min nO

M= κ/(4ωM ) 1. Ground-state

cooling becomes possible for high-frequency cantilevers (and/or high-finesse cav-

ities), when κ ωM . Then, we find

min nO

M ≈

κ

4ωM

2

. (16)

As explained in Ref. [33], these two regimes can be brought directly into cor-

respondence with the known regimes for laser-cooling of harmonically bound

atoms, namely the Doppler limit for ωM κ and the resolved sideband regime

for ωM κ.

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 mas-

ter equation, employing the rates obtained from the quantum noise approach.

However, as the coupling becomes stronger (e.g. by increasing the laser input

power), Γopt may reach the cavity decay rate κ. Then, the spectrum of force

fluctuations is itself modified by the presence of the cantilever. It becomes nec-

essary to solve for the coupled dynamics of the light field and the mechanical

motion. This has been done in Ref. [32], by writing down the Heisenberg equa-

tions of motion for the cantilever and the optical mode, and solving them after

linearization. Here we will only discuss the result.

7

radiation pressureforce

optical cavity

cantilever

laser

(a) (b) (c)

Figure 1: (a) The standard optomechanical setup treated in the text: A driven

optical cavity with a 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)

Radiation pressure force noise spectrum.

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 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 transis-

tors and microwave cavities coupled to nanobeams [20, 21, 22, 23, 24, 25, 26] to

clouds of 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 quan-

tum effects in these systems, including “cat” states [29], entanglement [30, 31]

and Fock state detection [12].

All the intrinsic optomechanical cooling experiments are based on the fact

that the radiation field introduces extra damping for the cantilever. In such a

classical picture, the effective temperature of the single mechanical mode of in-

terest is related to the bath temperature T by Teff/T = ΓM/(Γopt +ΓM ), where

ΓM and Γopt are the intrinsic mechanical damping rate and the optomechanical

cooling rate, respectively. Thus there is no limit to cooling in this regime, pro-

vided the laser power (and thus the cooling rate Γopt) can be increased without

any deleterious effects such as unwanted heating by absorption, and provided

the cooling rate remains sufficiently smaller than the mechanical frequency and

the cavity ring-down rate. However, at sufficiently low temperatures, the un-

avoidable 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 Ref. [32], based on the general quantum noise

approach. Independently, a derivation emphasizing the analogy to ion sideband

cooling was developed in Ref. [33]. In the present paper, we will review and

illustrate our theory. We start by outlining the basic classical picture, then

present the quantum noise approach that provides a transparent and straight-

forward way to derive cooling rates and quantum limits for the phonon number.

Finally, we illustrate the strong coupling regime that was first predicted in our

Ref. [32].

2

Calculate the force (i.e. shot-noise) spectrum


















sity:

SFF (ω) = dt eiωt

F (t)F (0)

. (6)










Γopt =x

2ZPF

2[SFF (ωM )− SFF (−ωM )] . (7)




Γopt↓ =

x2ZPF

2SFF (ωM ) , Γopt

↑ =x

2ZPF

2SFF (−ωM ). (8)




˙ρ =

(Γopt


†]

ρ (9)

5

figure by F. Marquardt

∆ = −ωM

F =ωR

La†awith

Clerk et al., RMP 82, 1155 (2010)

4 2 0 2 40

0.2

0.4

0.6

0.8

1

!/"

SF

F(!

)

∆ = +κ/2

Fano resonance in the force spectrum

Elste et al., PRL 102, 207209 (2009)

F = F1 + F2

F2 ∝ d+ d†F1 ∝ din + d†in

Linear theorywith two noise sources!

They are random, but not independent! d(ω) ∝ χR(ω)din

Zero temperature bath!

Γ↑opt ∝ SFF (−ωM )

This enables ground state cooling outside the resolved-sideband limit!SFF (ω) ∝

1−κ2+ i∆

χR(ω)

2

→Fano line shapeχR(ω) = [κ/2− i(ω +∆)]−1with

with

Fano resonance for dissipative coupling

Weiss, Bruder, and Nunnenkamp, arXiv:1211.7029also: Tarabrin et al., arXiv:1212.6242

Γopt

κ

For dissipative coupling there are two cooling regions cooling and two instability regions.

2 1 0 1 20.001

0.01

0.1

1

10

100

Ωmn osc

b

optΩm

nM

unst

able

unst

able

Strong-coupling effects in dissipatively coupled optomechanical systems 12

Figure 4. (a) Logarithm of the mechanical spectrum Scc(ω)κ as a function of detuning

∆. Parameters are ωm/κ = 3, ωm/γ = 105, nth = 100, A = 0, and Ba = 0.4. The

dark regions indicate regions of instability obtained from a numerical calculation. The

green curve gives half of the total damping rate, γtot/2, obtained from the quantum noise

approach with the origin shifted to (−2.5, 0). The dashed lines show the real parts of the

eigenvalues of the Hamiltonian (17). (b) Scc(ω)κ for detunings ∆/ωm = 1.05, 1, 0.95(from top to bottom). (c) Scc(ω)κ for detunings ∆/ωm = −0.9,−1,−1.1,−1.2,−1.3(from top to bottom).

purely dissipative and both types of coupling but has the same form as found in the case

of dispersive coupling only [5]. For B = 0 the result coincides with [5]; setting A = 0 the

result coincides with [14].

Figure 4 (a) shows the mechanical spectrum for strong dissipative coupling. Dark

areas indicate regions where the solutions of the linearized equations of motion are

unstable. This was numerically tested for the parameters used in Fig. 4 and coincides

with the regions where the total damping rate γtot from the weak-coupling approach is

negative. Whereas dispersive coupling leads to one unstable region for blue detuning,

dissipative coupling can lead to a second unstable region for red detuning in addition to

an unstable region for blue detuning. A third unstable region exists for even stronger drive

Normal-mode splitting in strong-coupling regime

Scc(ω) =

dt eiωtc†(t)c(0)

Normal-mode splitting

Regions of instability

Fano interference

∆ = −ωM

Weiss, Bruder, and Nunnenkamp, arXiv:1211.7029

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