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Annalen der Physik, 16 October 2012
A short walk through quantum optomechanicsP. Meystre
This paper gives an brief review of the basic physics ofquantum
optomechanics and provides an overview ofsome of its recent
developments and current areas of fo-cus. It first outlines the
basic theory of cavity optomechan-ical cooling and gives a brief
status report of the exper-imental state-of-the-art. It then turns
to the deep quan-tum regime of operation of optomechanical
oscillatorsand cover selected aspects of quantum state prepara-
tion, control and characterization, including
mechanicalsqueezing and pulsed optomechanics. This is followed bya
discussion of the bottom-up approach that exploits ul-tracold
atomic samples instead of nanoscale systems. Itconcludes with an
outlook that concentrates largely on thefunctionalization of
quantum optomechanical systems andtheir promise in metrology
applications.
1 Introduction
Broadly speaking, quantum optomechanics provides auniversal tool
to achieve the quantum control of mechan-ical motion [1]. It does
that in devices spanning a vastrange of parameters, with mechanical
frequencies from afew Hertz to GHz, and with masses from 1020g to
severalkilos. At a fundamental level, it offers a route to
deter-mine and control the quantum state of truly
macroscopicobjects and paves the way to experiments that may leadto
a more profound understanding of quantum mechan-ics; and from the
point of view of applications, quantumoptomechanical techniques in
both the optical and mi-crowave regimes will provide motion and
force detectionnear the fundamental limit imposed by quantum
mechan-ics.
While many of the underlying ideas of quantum op-tomechanics can
be traced back to the study of gravita-tional wave detectors in the
1970s and 1980s [2, 3], thespectacular developments of the last few
years rely largelyon two additional developments: From the top
down,it is the availability of advanced micromechanical
andnanomechanical devices capable of probing extremelytiny forces,
often with spatial resolution at the atomicscale. And from the
bottom-up, we have gained a detailedunderstanding of the mechanical
effects of light and howthey can be exploited in laser trapping and
cooling. Thesedevelopments open a path to the realization of
macro-scopic mechanical systems that operate deep in the quan-tum
regime, with no significant thermal noise remaining.
As a result, they offer both knowledge and control of thequantum
state of a macroscopic object, and increasedsensitivity, precision,
and accuracy in the measurementof feeble forces and fields.
It was Arthur Ashkin [4] who first suggested anddemonstrated
that small dielectric balls can be acceler-ated and trapped using
the radiation-pressure forces as-sociated with focused laser beams.
In later experimentsthese particles, weighting on the order of a
microgram,were levitated against the Earth gravitational field.
This ad-vance led to the realization of optical tweezers, whose
ap-plications in biological science have become ubiquitous.In
parallel, the use of the strong enhancement providedby resonant
light scattering lead to the laser cooling ofions and of neutral
atoms by D. Wineland, T. W. Hnsch, S.Chu, W. D. Phillips, C.
Cohen-Tannoudji and many others,resulting in a wealth of
extraordinary developments [5]culminating in 1995 with the
invention of atomic Bose-Einstein condensates [6, 7], and the
subsequent explosionin the study of quantum-degenerate atomic
systems.
Non-resonant light-matter interactions present theconsiderable
advantage of being largely wavelength in-dependent, providing one
with the potential to achieveoptomechanical effects for a broad
range of wavelengthsfrom the microwave to the optical regime.
Resonant in-teractions, on the other hand, can result in a very
largeenhancement of the interaction, but at the cost of be-ing
limited to narrow ranges of wavelengths. Cavity op-tomechanics
exploits the best of both worlds by achievingresonant enhancement
through an engineered resonant
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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 witha series of
narrow resonances, or an electromagnetic res-onator such as a
superconducting LC circuit. Indeed, nu-merous designs can achieve
optomechanical control viaradiation pressure effects in
high-quality resonators. Theyrange from nanometer-sized devices of
as little as 107
atoms to micromechanical structures of 1014 atoms andto
macroscopic centimeter-sized mirrors used in gravita-tional wave
detectors.
That development first appeared at the horizon in the1960s, but
more so in the late 1970s and 1980s. It wasinitially largely driven
by the developments in opticalgravitational 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 theend-mirrors of a large path length optical
interferometer.Changes in the optical path length due to local
changesin the curvature of space-time modulate the frequency ofthe
cavity resonances and in turn, modulate the opticaltransmission
through the interferometer. It is in this con-text that researchers
understood fundamental quantumoptical effects on mechanics and
mechanical detectionsuch as the standard quantum limit, and how the
basiclight-matter interaction can generate non-classical statesof
light.
Braginsky and colleagues demonstrated cavity op-tomechanical
effects with microwaves [9] as early as 1967.In the optical regime,
the first demonstration of theseeffects was the radiation-pressure
induced optical bista-bility in the transmission of a Fabry-Prot
interferometer,realized by Dorsel el al. in 1983 [10]. In addition
to theseadiabatic effects, cooling or heating of the mechanical
mo-tion is also possible, due to the finite time delay betweenthe
mechanical motion and the response of the intracavityfield, see
section 2.2. The cooling effect was first observedin the microwave
domain by Blair et al. [11] in a Niobiumhigh-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 ofa mechanical mirror by Cohadon et al. [12],
followed byphotothermal cooling by Karrai and coworkers [13],
andshortly thereafter by radiation pressure cooling in
severalgroups [1419]. Also worth mentioning is that as early as1998
Ritsch and coworkers proposed a related scheme tocool atoms inside
a cavity [20].
This paper reviews the basic physics of quantum op-tomechanics
and gives a brief overview of some of itsrecent developments and
current areas of focus. Section2 outlines the basic theory of
cavity optomechanical cool-ing and sketches a brief status report
of the experimentalstate-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 stepin
quantum optomechanics. Quantum state preparation,control and
characterization are the next challenges ofthe field. Section 3
gives an overview of some of the majortrends in this area, and
discusses topics of much currentinterest such as the so-called
strong-coupling regime, me-chanical squeezing, and pulsed
optomechanics. Section 4discusses a complementary bottom-up
approach thatexploits ultracold atomic samples instead of
nanoscalesystems to study quantum optomechanical effects.
Finally,Section 5 is an outlook that concentrates largely on
thefunctionalization of quantum optomechanical systemsand their
promise in metrology applications.
2 Basic theory
To describe the basic physics underlying the main aspectsof
cavity optomechanics it is sufficient to consider an op-tically
driven Fabry-Prot resonator with one end mirrorfixed -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 L
, see Fig. 1. Braginskyrecognized as early as 1967 [9] that as
radiation pressuredrives the mirror, it changes the cavity length,
and hencethe intracavity light field intensity and phase. This
resultsin two main effects: the optical spring effect, an
opti-cally induced change in the oscillation frequency of themirror
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 fluidthat can damp the mirror
motion and cool its center-of-mass motion.
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Ann. Phys. (Berlin) 0, No. 0 (2012)
One can immediately understand how the opticalspring effect can
result in a more quantum behavior of theoscillator by recalling
that in the high temperature limitthe mean number of phonons nm in
the center-of-massmotion of an oscillator of frequencym is given
by
nm = kBT /m , (1)where kB is Boltzmans constant and T the
temperature.For a given temperature, increasing m
automaticallyreduces nm, allowing one to approach the quantumregime
without having to reduce the temperature.
Cold damping, in contrast, does reduce the tempera-ture of the
oscillating mirror by opening up a dissipationchannel to a
reservoir that is effectively at zero temper-ature. To see how this
work, we first remark that in theabsence of optical field the
oscillating mirror is dissipa-tively coupled to a thermal bath at
temperature T . Its av-erage center-of-mass energy, E, results from
the balancebetween dissipation and heating,
dEdt
=E+kBT, (2)
where is the intrinsic mechanical damping rate. Whenan optical
field is applied, an additional optomechanicaldamping channel with
damping rate opt comes into playso that
dEdt
=E+kBT optE. (3)
Importantly, that channel does not come with an addi-tional
(classical) thermal bath. Optical frequencies aremuch higher than
mechanical frequencies, so that the op-tical field is effectively
coupled to a reservoir at zero tem-perature. In steady state Eq.
(3) gives E = kBT /(+opt),or
Teff =T
+opt. (4)
This simple phenomenological classical picture predictsthat the
fundamental limit of cooling is T = 0. A moredetailed quantum
mechanical analysis does yield a funda-mental limit given by
quantum noise, see Section IIC, butin practice, this is usually not
a major limitation to cool-ing the mechanical mode arbitrarily
close to the quantumground state, nm = 0.
More quantitatively, we consider a single mode of theoptical
resonator of nominal frequency c and assumethat radiation pressure
results in a displacement x(t) ofthe harmonically bound end-mirror,
and consequently ina change in the optical mode resonance frequency
to
c =c Gx(t ), (5)
where
G =c/x. (6)
For a single-mode Fabry-Prot resonator of length L thisbecomes
simply G =c/L.
Typical mechanical oscillator frequencies are in therange
ofm/2pi=10Hz to 109Hz and the mechanical qual-ity factors of the
mirrors are in the range of perhaps Qm 103107, so that typically
the damping rate =m/Qmof the oscillating mirror is much slower than
the damp-ing rate of the intracavity field. One can then gain
con-siderable intuition by first neglecting mirror damping
al-together and assuming that its motion is
approximatelyharmonic,
x(t ) x0 sin(m t ). (7)
For a classical monochromatic pump of frequencyL andamplitude in
the intracavity field obeys the equation ofmotion
d(t )
dt= [i (+Gx(t ))/2](t )+pin, (8)
with the steady-state solution
=pin
i (+Gx)+/2 . (9)
Here we have introduced the detuning
=L c (10)
and is the intracavity field amplitude, normalized sothat
||2 = (+Gx)2+ (/2)2
(P
L
)=
(+cx/L)2+ (/2)2(
P
L
)(11)
where
P =L |in|2 (12)
is the input laser power driving the cavity mode. This
nor-malization allows for an easy generalization to the case
ofquantized fields, in which casewill be interpreted as thesquare
root of the mean number of intracavity photons,
=aa, with a and a the bosonic annihilation and
creation operators of the intracavity field. Note that |in|2has
then the units of photons per second.
For oscillation amplitudes x0 small enough thatGx0/m 1, it can
be shown that the mirror oscillations simply re-sult in the
generation of two sidebands at frequencies
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P. Meystre: A short walk through quantum optomechanics
L m , see e.g. Refs. [21, 22]. The time-dependent com-plex field
amplitude(t ) then takes the approximate form(t )'0(t )+1(t )
with
0(t ) 'pin
i+/2 , (13)
1(t ) '(Gx0
2
) pin
i+/2 (14)
(
eim t
i (+m)+/2 e
+im t
i (m)+/2)
.
The first sideband in Eq. (15) can be interpreted as an
anti-Stokes line, with a resonance at L = c m , and thesecond one
is a Stokes line. An important feature of thesesidebands is that
their amplitudes can be vastly different,as they are determined by
the cavity Lorentzian responsefunction evaluated at L m and L +m ,
respectively.
2.1 Static phenomena
Consider first a situation where the cavity damping rate is much
faster than all other characteristic times of thesystem. One can
then understand the mirror motion as re-sulting from the combined
effects of the harmonic restor-ing force and the radiation pressure
force Frp resultingfrom an adiabatic elimination of the intracavity
field, seee.g. Ref. [23],
Frp =G||2 =cL||2, (15)
where ||2 is given by Eq. (11) and the second equalityholds for
a simple Fabry-Prot. One can easily show thatthe force Frp can be
derived from the potential
Vrp =||2
2arctan[2(+Gx)/] , (16)
the mirror of mass m being therefore subject to the
totalpotential
V (x)= 12m2mx
2 ||2
2arctan[2(+Gx)/] . (17)
The potential Vrp slightly displaces the equilibrium po-sition
of the mirror to a position x0 6= 0, as would be in-tuitively
expected, and also changes its spring constantfrom its intrinsic
value k =m2m to a new value
krp =m2m +d2Vrp(x)
dx2
x=x0
. (18)
The second term in this expression is the static opticalspring
effect. For realistic parameters it can increase the
stiffness of the mechanical system by orders of magnitude.A
third important static effect of radiation pressure is thatin
general, there is a range of parameters for which thepotential V
(x) can exhibit 3 extrema. Two of them corre-spond to stable local
minima of V (x), and the third one toan unstable maximum. This
results in radiation pressureinduced optical bistability [10], an
effect that is physicallysimilar to the more familiar form of
bistability that canoccur in a Kerr nonlinear medium. The
difference is thatin one case, it is the optical length of the
resonator thatis changed by a Kerr nonlinearity, with its physical
lengthremaining unchanged, while in the other it is that
physicallength that is intensity-dependent.
2.2 Effects of retardation
In general the optical field does not respond instantly tothe
motion of the mechanical oscillator, therefore we needto account
for the effects of retardation as well. We pro-ceed by assuming
that he system is in equilibrium at somemirror position x0 with
intracavity field 0, taken to bereal without loss of generality,
and consider the linearizeddynamics of small displacements x(t) and
(t) fromthat state under the effect of an external force F (t
),
x+x+2mx = G0(+) ,
= (i/2)+ iG0x. (19)These equations of motion can easily be
solved, for in-stance in Fourier space, to give
()=(
iG0i (+)+/2
)x() (20)
where
=+Gx0, (21)resulting in a modification of the radiation pressure
force
Frp()=G0[()+()] . (22)
Together with Eq. (20) this expression shows that the mir-ror
motion exerts a dynamical back-action on the radi-ation pressure
force, which acquires both a real and animaginary component, the
physical origin of the imagi-nary component being the delayed
response of the intra-cavity field. As a result the intracavity
power acquires acomponent that oscillates out of phase with the
mirrormotion, that is, with its velocity. It is through that
frictionforce that the optical field acts as a viscous field for
themirror.
The net effect of the real and imaginary components ofFrp can be
conveniently cast in terms of the back-action
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Ann. Phys. (Berlin) 0, No. 0 (2012)
frequency shift opt and a damping rate opt, see e.g.Ref. [21,
22], with
opt =G2202mm
[+m
(+m)2+2/4+ +m
(m)2+2/4]
(23)
and
opt =G2202mm
[
(+m)2+2/4
(m)2+2/4]
.
(24)
For detunings m the first term in Eq. (24) dom-inates over the
second term, and the dynamical back-action results in an increase
in the mechanical dampingof the mechanical oscillator and cooling,
see Eq. (4). Itis therefore the asymmetry between the response
func-tion of the Fabry-Prot at the frequencies of the two sidemodes
that is responsible for cooling or anti-dampingif one changes the
sign of and uses a blue-detuned in-stead of a red-detuned driving
field. In particular, in theresolved sideband limitm we find
opt (
2
) G220mm
, (25)
which can in principle be increased arbitrarily (within
thelimits of validity of the model) by increasing the
incidentoptical power.
Together with Eq. (4) this analysis predicts that thecooling of
the center-of-mass motion of the mirror canbe arbitrarily close to
Teff = 0, a consequence of the factthat the optomechanical coupling
between the intracav-ity field and the mirror results in the
scattering of thethe driving field into an anti-Stokes line that is
stronglydamped due to the high density of states at the cavity
res-onance. Conversely, for the opposite detuning mit is the Stokes
line that is strongly damped, resulting inanti-damping of the
mirror motion. This can lead to para-metric oscillations and
dynamical instabilities, a situationfurther discussed in section
3.5.
The quantum description of the next section will showthat cold
damping and mirror cooling can also be inter-preted in terms of of
the annihilation of phonons from thecenter-of-mass mode of
oscillation when scattering thedriving laser field into the
anti-Stokes sideband. Heatingcan similarly be understood as
resulting from the creationof phonons associated with the
scattering of the drivingfield into the lower frequency Stokes side
mode.
Figure 2 (Color online) Schematic of sideband cooling: a
co-herent light field driving the resonator acquires frequency
side-bands due to the mirror oscillations. The origin of the
highfrequency sideband is the parametric transfer of phonons
fromthe mirror to the optical field and the lower sideband is due
tothe reverse process, see section 2.2. Sideband cooling
resultswhen the upper sideband frequency is resonant with the
res-onator. The solid black curve depicts the resonator
transmissionnear its mode of frequency c .
2.3 Quantum limit
The classical prediction that one can in principle reach
anarbitrarily large degree of cooling needs to be revised toaccount
for the effects of quantum and thermal noise. Asis well known, the
open port of the interferometer usedto supply the optical drive of
the oscillating mirror alsoallows for the coupling of vacuum
fluctuations into theresonator, see e.g. Ref. [24]. This leads to a
fundamen-tal limit to the degree of cooling that can be achieved.
Aproper quantum description of the system must accountfor this
effect as well as for the the bosonic nature of thephonons.
Ignoring in a first step the important effects of fluctua-tions
and dissipation, and in case a single optical modeof the Fabry-Prot
resonator and a single mode of oscilla-tion of the suspended mirror
need to be considered, theoptomechanical Hamiltonian is simply
H =(x)aa+ p2
2m+ 1
2m2m x
2, (26)
where a and a are bosonic annihilation and creation op-erators
for the cavity mode of frequency, and p and x arethe momentum and
position of the oscillating mirror ofmass m and frequencym . In
reality, though, this Hamil-tonian is more subtle than may appear
at first. This isbecause the mode frequency (q) depends on the
lengthof the resonator, which in turn depends on the
intracavityintensity. Stated differently, the boundary conditions
forthe quantization of the light field are changing in time,and do
so in a fashion that depends on the state of that
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P. Meystre: A short walk through quantum optomechanics
field and its history. The rigorous quantization of this sys-tem
is a far-from-trivial problem, but for most cases ofinterest in
quantum optomechanics the situation is signif-icantly simplified
since the transit time c/2L of the lightfield through the optical
resonator is much faster than themechanical frequencym . The
intracavity field thereforelearns about changes in its environment
in times shortcompared to 1/m . Under these conditions one can
as-sume that the cavity frequency follows adiabatically anychange
in resonator length,
(x)= npicL+ x =c
(1
1+ x/L)c (1 x/L) (27)
where n is an integer that labels the mode of nominalfrequencyc
and L is the nominal resonator length (in theabsence of light.) In
the classical limit we recover the resultG =c/L valid for a simple
Fabry-Prot. The Hamiltonian(26) reduces then to [2527]
H = c aa+ p2
2m+ 1
2m2m x
2Gaax (28)
= c aa+ p2
2m+ 1
2m2m x
2g0aa(b+ b).In the second line we have used the familiar
relationshipbetween the position operator x and the annihilation
andcreation operators b and b of the mechanical oscillator,
x = xzpt(b+ b) (29)with
xzpt =
2mm
. (30)
We also introduced the optomechanical coupling fre-quency
g0 = xzpfG =xzpfc/x, (31)which scales the optomechanical
displacement to thezero-point motion of the mechanical oscillator.
TheHamiltonian (28) is the starting point for most
quantummechanical discussions of cavity optomechanics.
In order to establish the theoretical limit to cavity
op-tomechanical cooling, it is necessary to expand the de-scription
provided by the Hamiltonian (28) to account forthe optical drive of
the resonator, cavity damping, and themechanical damping of the
oscillator. This analysis wascarried out in Refs. [2830]. The main
message of thesepapers is that at least for constant optomechanical
cou-pling the best cooling can be achieved in the so-called
re-solved sideband limit, m , with the minimum meanphonon
number
nm =optn0m +nTm
+opt. (32)
Here n0m is the mean steady-state number of phonons inthe
absence of mechanical damping, given by the detailedbalance
expression
n0m +1n0m
= (+m)2+2/4
(m)2+2/4 exp
( mkBTeff
), (33)
nTm is the equilibrium phonon occupation determined bythe
mechanical bath temperature, and
opt =g 20aa
2mm(34)
[
(+m)2+2/4
(m)2+2/4]
.
For nTm 0 one recovers the classical result of Eq. (4).If the
optical damping opt dominates, opt , though,the mean phonon number
is limited in the resolved side-band limitm to
n0m =(
4m
)2, (35)
which shows that the ground state can be approached, butnot
reached, in that case. As expected from the classicalconsiderations
of the preceding section, this is the bestpossible case. In
practice, the theoretical limit (35) is diffi-cult to reach due to
technical noise issues including lasernoise [31, 32], clamping
noise [33], etc. but the discussionof these topics in beyond the
scope of this brief review.Remarkably though, these experimental
challenges havenow being overcome in several experiments, see
section2.4. We also note that using pulsed optomechanical
inter-actions may lead to improved cooling limits [34, 35].
Wereturn briefly to this point in section 3.7.
Importantly, we remark that optomechanical sidebandcooling is
formally identical to the cooling of harmoni-cally trapped ions, or
more generally of any harmonicallytrapped dipole, see Ref. [36] for
a nice discussion of thispoint. In the case of trapped ions, the
resolved sidebandcooling limit was understood as early as 1975 [37,
38], andthe ground state cooling of trapped ions was first
demon-strated over 20 years ago [39, 40]. As already mentioned,the
key new element contributed by cavity optomechan-ics is the use of
engineered resonance-enhancing struc-tures.
2.4 Experimental status
Following the pioneering work on gravitational wave an-tennas,
advances in material science and nanofabrica-tion in particular in
microelectromechanical systems(MEMS), nanoelectromechanical systems
(NEMS), and
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Ann. Phys. (Berlin) 0, No. 0 (2012)
Figure 3 (Color online) Artist conception of the
microwaveoptomechanical circuit of Ref. [42]. Capacitor element of
theLC circuit is formed by a 15 micrometer diameter
membranelithographically suspended 50 nanometers above a lower
elec-trode. Insert: cut through the capacitor showing the
membraneoscillations. After Ref. [44].
ba
0.1
1
10
100
Occ
upan
cy
100 101 102 103 104 105
Drive Photons, nd
nm nc
10-32
10-31
10-30
10-29
S x (
m2 /
Hz)
10.55810.556Frequency (M Hz)c
n =27m
n =22m
n =8.5m
n =2.9m
n =0.93m
n =18d
n =71d
n =280d
n =1,100d
n =4,500 d
10-15
2
4
68
10-14
2
4
10-13
S/P o
(1/
Hz)
10.710.610.510.410.3Frequency (M Hz)
n =0.93m
n =0.55m
n =0.36m
n =0.34m
n =0.42mg /
n =4,500 d
n =11,000d
n =28,000d
n =89,000d
n =180,000d
Figure 4 (Color online) Phonon occupancy (blue) and intra-cavity
photon occupancy (red) as a function of the drive photonnumber. In
this example sideband cooling reduces the thermaloccupancy of the
mechanical mode from nm=40 into the quan-tum regime, reaching a
minimum of nm=0.34 0.05. FromRef. [42], with permission.
optical microcavities opened up the possibility to ex-tend these
ideas in many new directions, leading to thedemonstration of
significant cooling in a broad varietyof systems from 2006 on, see
Refs. [1518], with the firstdemonstration of cooling in the
resolved sideband regimereported in Ref. [36].
More recently these efforts have culminated in thecooling of the
center-of-mass motion of at least three dif-ferent micromechanical
systems with a mean phononnumber within a fraction of a phonon of
their groundstate of vibrational motion, nm < 1 [4143]. We
post-pone a discussion of Ref. [41] until the next section
toconcentrate first on the two experiments [42, 43] thatutilized
resolved sideband cooling to approach the me-
chanical ground state of center-of-mass motion. In onecase [42]
the mechanical resonator was a suspended cir-cular aluminum
membrane tightly coupled to a super-conducting lithographic
microwave cavity. That cavitywas precooled to 20mK, corresponding
to an initial occu-pation of 40 phonons and then further cooled by
radia-tion pressure forces to an average phonon occupation ofnm
0.3. In contrast, Ref. [43] utilized an optomechan-ical structure
with co-located photonic and phononicband gaps in a suspended
on-chip waveguide. The struc-ture was precooled to 20K,
corresponding to about 100thermal quanta, and then cooled via
radiation pressureto nm 0.85. Shortly thereafter, that same group
alsoobserved the motional sidebands generated on a secondprobe
laser by a mechanical resonator cooled opticallyto near its
vibrational ground state. They were able to de-tect the asymmetry
in the sideband amplitudes betweenup-converted and down-converted
photons, a smokinggun signature of the asymmetry between the
quantumprocesses of emission and absorption of phonons [45].
3 Beyond the ground state
3.1 Strong coupling regime
Cooling mechanical resonators to their ground state ofmotion is
an essential first step in eliminating the ther-mal fluctuations
that normally mask quantum features.However, by itself that state
is not particularly interest-ing, so the next challenge is to
prepare, manipulate andcharacterize quantum states of the
mechanical resonatorrequired for a specific science or engineering
goal. Animportant first experimental step in that direction
wasreported in Ref. [41]. In contrast to Refs. [42] and [43]this
experiment did not rely on radiation pressure cool-ing to achieve
the motional ground state. Because of itshigh frequency of about 6
GHZ, a conventional dilutionrefrigerator that can reach
temperatures of about 25 mKwas sufficient to cool it to nm <
0.07. A key point of theexperiment is that it succeeded in coupling
an acousticresonator to a two-state system, or qubit, that could
detectthe presence of a single mechanical phonon. This is
analo-gous to protocols that have been developed over the yearsin
cavity quantum electrodynamics, see e.g. Ref. [46], withthe
important distinction that photons are now replacedby phonons.
The coupling between a bosonic field mode and oneor more
two-state systems paves the way to a numberof approaches to prepare
and to observe genuine quan-tum features such as the energy
quantization of the res-
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P. Meystre: A short walk through quantum optomechanics
onator, or to make controlled state manipulations at thefew
phonons level. Many of those protocols have alreadybeen developed
in quantum optics and can be readily ap-plied to phonon fields, at
least in principle. In all casesdissipation and decoherence must be
reduced to a mini-mum, as they rapidly lead to the destruction of
the mostsalient quantum features of the state. In the experiment
ofRef. [41] decoherence was just weak enough to observe afew
coherent oscillations of a single quantum exchangedbetween the
qubit and the mechanical structure. As suchit can be considered as
the first demonstration of the capa-bility of coherent control of
phonon fields in a microme-chanical resonator.
Generally speaking, and in complete analogy with thesituation in
quantum optics and in cavity QED, the con-trol of the quantum state
of a mechanical oscillator re-quires that one operates in the
so-called strong couplingregime, where the energy exchange between
the mechan-ical object and the system to which it is coupled
anoptical field mode, a qubit, an electron, etc. is not nega-tively
affected by dissipation and decoherence. Section 2.3showed that at
the simplest level the optomechanical in-teraction takes the form
(28),
H =c aa+ p2
2m+ 1
2m2m x
2g0aa(b+ b). (36)
At the single photon level, aa = 1, this interaction isusually
much too weak for its coherent nature to domi-nate over the
incoherent dynamics for realizable levels ofdecay and decoherence.
Since for aa 1 the quantumnature of the optical field normally
rapidly decreases inimportance, it is therefore challenging to
reach situationswhere the full quantum nature of the interaction
betweenthe photon and phonon fields is significant. There is a
wayaround this difficulty, though, the trade-off being that
theintrinsic nonlinear nature of the optomechanical inter-action
(36) disappears in the process to be replaced by alinear effective
interaction. As we shall see, this is not allbad, as that effective
interaction offers itself a number ofnew opportunities.
Our starting point is the observation that strong in-tracavity
optical fields can usually be decomposed as thesum of a classical,
or mean-field part and a small quan-tum mechanical component. In
terms of the mean fieldof the optical field mode = aa+ c (37)where
c is again a photon annihilation operator. The op-tomechanical
coupling term in the Hamiltonian (36) be-comes then
Hint =g0n(b+ b)g(c+ c
)(b+ b) (38)
where we have introduced the optomechanical couplingstrength
g = g0pn, (39)
n = ||2, and we have taken to be real for notational
con-venience. The first term in the Hamiltonian (38) describesa
simple Kerr effect, with a change in resonator length pro-portional
to the classically intracavity intensity. This is theterm that
leads to the radiation pressure induced opticalbistability observed
e.g. in the experiments of Dorsel etal [10].
In a frame rotating at the driving field frequency, thecavity
frequency and the mechanical frequency the sec-ond term in Eq. (38)
can be reexpressed as
V = g[bcei (+m )t +h.c.
] g
[bcei (m )t +h.c.
](40)
This interaction describes the linear coupling betweenthe
quantized component of the optical field and the me-chanical
oscillator. The coupling g is enhanced from thesingle-photon
optomechanical coupling frequency g0 bya factor
pn, which can be very substantial. Note however
that this enhancement comes at the cost of losing the non-linear
character of the original interaction g0aa(b+ b).That nonlinear
character is at the origin of a number ofquantum effects that are
expected to appear when theradiation pressure of a single (or of
very few) photonsdisplaces the mechanical oscillator by more than
xzpf.These include two-photon blockade as well as quantita-tive
changes in the output spectrum and cavity responseof the
optomechanical system, leading for example to thepossible
generation of non-Gaussian steady states of theoscillator
[4749].
The linear coupling of Eq. (40) provides exciting oppor-tunities
as well, and these are significantly less challengingto realize
experimentally. On the red-detuned side of theFabry-Prot resonance,
=m , we have after invokingthe rotating wave approximation
V 'g(bc+h.c.
), (41)
the so-called beam-splitter Hamiltonian of quantum op-tics. In
contrast, in the blue-detuned side of the resonance,=+m , we haveV
'g
(bc+h.c.
), (42)
which describes the parametric amplification of thephonon mode
and the optical field.
This approach has enabled experiments to reach theregime of
strong phonon-photon optomechanical cou-pling in several
micromechanical devices [5052]. A fa-miliar characteristic of
strongly coupled systems is the
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Ann. Phys. (Berlin) 0, No. 0 (2012)
occurrence of normal mode splitting. For the Hamilto-nian (40)
the normal mode frequencies are
= 12
[2+2m
(22m
)2+4g 2m]1/2 . (43)The first demonstration of normal mode
coupling in anoptomechanical situation was realized by Grblacherand
coworkers [50]. As pointed out by these authors theoptomechanical
modes can be interpreted in a dressedstate approach as excitations
of mechanical states thatare dressed by the cavity radiation field.
Alternatively,they can also be interpreted as optomechanical
polari-ton modes. Teufel and coworkers [51] carried a seriesof
experiments in the strong coupling regime of quan-tum
optomechanics. They measured the dressed cavitystates as a function
of the pump-probe experiment wherethe coupling (39) was controlled
by a pump field andthe resonator transmission measured by a weak
probefield. Increasing the strength of g allowed them to mon-itor
the change in cavity transmission as the strong cou-pling regime
was reached, with an intermediate regimewhere the interference
between the pump and probe fieldresults in an effect analogous to
electromagnetically in-duced transparenty [53, 54].
It should be emphasized that by itself, the observationof normal
mode splitting, which is both a classical andquantum feature of
coupled systems, does not prove theexistence of coherent exchange
of excitations between themechanical and optical field modes. An
important steptoward the demonstration of quantum coherent
couplingwas recently achieved by Kippenberg and coworkers [52]in a
system where the optomechanical coupling is de-scribed by the beam
splitter interaction (41). This exper-iment considered a
micro-mechanical oscillator cooledto a mean phonon number of the
order of nm 1.7, andin addition excited the system with a weak
classical lightpulses to achieve coherent coupling between the
opticalfield and the micromechanical oscillator and the level
ofless than one quantum on average. These results, whilestill
preliminary in many ways, open up a promising routetowards the use
of mechanical oscillators as quantumtransducers, as well as in
microwave-to-optical quantumlinks as we now discuss.
3.2 State transfer
The beam-splitter Hamiltonian (41) describes the coher-ent
exchange of cavity photons and mechanical phonons.One of its
remarkable properties is that it offers the poten-tial to precisely
transfer the quantum state of the mechan-ical oscillator to the
electromagnetic field, and conversely.
0
1
Pro
be tr
ansm
issi
on, |
T|2
7.47407.47357.47307.4725Probe frequency,
102
103
104
105
106
100 102 104 106
7.4740
7.4735
7.4730
7.4725
a
c
bnd=5x10
-1
/2 (GHz) p
/
2 (G
Hz)
p g
/ (H
z)
n d
nd=5x100
nd=5x101
nd=5x102
nd=5x103
nd=5x104
nd=5x105
nd=5x106
Cou
plin
g ra
te,
g /
Drive photons,
Figure 5 (Color online) Normalized cavity transmission
forincreasing resonator drive intensity nd = ||2. For moderatedrive
intensities the interference between the drive and probephotons
results in a narrow peak in the cavity spectrum, theonset of
electromechanically induced transparency. For higherintensities the
cavity resonance then splits into normal modes.From Ref. [51], with
permission.
This is seen easily by considering the Heisenberg equa-tions of
motion for the annihilation operators b and c inthe absence of
decay,
b(t ) = b(0)cos(g t )+ i c(0)sin(g t ),c(t ) = c(0)cos(g t )+ i
b(0)sin(g t ). (44)The optomechanical interaction g can easily be
madetime dependent by pulsing the classical driving laser
fieldintensity, n n(t ). For an interaction time tint and a
driv-ing laser pulse intensity such that
g0 tint
0dtn(t )t =pi/2
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P. Meystre: A short walk through quantum optomechanics
we then have that b(tint)= c(0) and c(tint)= i b(0), indica-tive
of a perfect state transfer between the optical andphonon modes
assuming of course that dissipation anddecoherence can be ignored
during that time interval.
The interest in devices capable of high-fidelity statetransfer
between optical and acoustical fields is largelymotivated by its
potential for quantum information ap-plications. This is because
due to their potentially slowdecoherence rate motional states of
mechanical systemsare well suited for information storage. However
mechan-ics does not permit fast information transfer, while
op-tical fields are ideal as information carriers, but are
typ-ically subject to fast decoherence that limits their inter-est
for storage [55]. The coherent quantum mapping ofphonon fields to
optical modes also promises to be use-ful in quantum sensing
applications, by combining theremarkable sensitivity of nanoscale
cantilevers to feebleforces and fields with reliable and
high-efficiency opti-cal detection schemes. And in addition to
standard statetransfer between motional and optical states,
phononfields could also serve as convenient transducers
betweenoptical fields of different wavelengths, or between
opticaland microwave fields.
The first theoretical proposal that analyzed a schemeto transfer
quantum states from a propagating light fieldto the vibrational
state of a movable mirror by exploit-ing radiation pressure effects
is due to Jin Zhang andcoworkers [56]. This work was then expanded
in severaldirections, especially in the context of quantum
optome-chanics. For instance Tian and Wang [57] proposed
anoptomechanical interface that converts quantum statesbetween
optical field of distinct wavelengths through asequence of
optomechanical pi/2 pulses. In another re-cent proposal, Didier et
al. [58] considered exploiting thebeam splitter coupling of a
mechanical oscillator and amicrowave resonator to measure and
synthesize quantumphonon states, and also to generate and detect
entangle-ment between phonons and photons. They also
proposedgenerating the entanglement of two mechanical oscilla-tors
and its detection by the cavity field after entangle-ment swapping.
The first experimental demonstration ofstate transfer between a
microwave field and a mechani-cal oscillator with amplitude at the
single quantum levelwas recently achieved by Palomaki et al.
[59].
3.3 Two-mode squeezing
The Hamiltonian (42) is essentially the familiar
two-modesqueezing Hamiltonian of quantum optics. This
becomesreadily apparent if one accounts for the (controllable)
phase of the classical driving field, so that
g = g0pn i g0
pn exp(i)
and
V =i[g bc gbc
](45)
with the associated evolution operator
Sab(t )= exp[(gbc g bc)t ], (46)the well-known unitary two-mode
squeezing operator. In-troducing the generalized two-mode
quadrature operator
Xab =1
23/2(c+ c+ b+ b) (47)
one finds that the variance of a system initially in a two-mode
vacuum state is given by [60]
(X )2 = 14
[e2|g |t cos2(/2)+e2|g |t sin2(/2)] . (48)
That same result also holds if the two modes are initiallyin
coherent states. For the choice = pi/2 one finds im-mediately that
(X )2 can be well below the standardquantum limit of 1/4, a
signature of two-mode squeez-ing. Two-mode squeezed states are
known to be entan-gled, indicating that this form of interaction
can result inquantum entanglement between the photon and
phononmodes. As such this configuration represents a useful
re-source for demonstrating fundamental quantum mechan-ical effects
as well as for exploiting cavity optomechanicaldevices in a quantum
information context.
We note for completeness that in early work, Fabreand coworkers
[61], and independently Mancini andTombesi [62] exploited the
analogy between the situa-tion of an optical resonator and a cavity
filled by a Kerrmedium to predict single mode squeezing of the
reflectedoptical field in situations where the motion of the
mirroris dominated by thermal fluctuations and can be
treatedclassically.
3.4 Squeezing via back-action evadingmeasurements
As shown by Braginsky et al. [63] and further analyzed byClerk
et al. [64] it is possible to implement back-actionevading
measurements of the membrane position whendriving it with an input
field resonant with the cavity fre-quency c , but modulated at the
mirror frequency m .The mean-field amplitude of the intracavity
field is then
(t )=pn cos(m t ), (49)
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Ann. Phys. (Berlin) 0, No. 0 (2012)
where
=pin
im +/2. (50)
Introducing the quadratures
X (t ) = 1p2
(ce im t + ceim t
),
Y (t ) = ip2
(ce im t ceim t
)(51)
of the motional mode, with [X (t ), Y (t )]= i andx(t
)=p2xzpt
(X (t )cosm t + Y (t )sinm t
), (52)
and keeping as before only linear terms in the quantumcomponent
of the field, the optomechanical interactionHamiltonian reduces
then to
V =p2g [X (1+cos(2m t ))+ Y sin(2m t )] (b+ b)(53)
where g = g0= g0pn as before.
In a time-averaged sense the interaction Hamilto-nian (53)
reduces to
V p2g X (b+ b) (54)and commutes with X , thus giving rise to the
possibil-ity of performing a back action evading measurement ofthe
X quadrature of mirror motion. This was verified ex-perimentally in
the classical regime by J. B. Hertzberg etal. [68], but not yet in
the quantum regime so far.
Since the interaction (54) is linear in X it is perhapsless
evident that it can also lead to quadrature squeezing.This can be
achieved by first performing a precise mea-surement of X ,
following which its quadrature can clearlybe below the standard
quantum limit. Following that mea-surement the system would
normally rapidly relax back toa classical state, but by applying an
appropriate feedback,the measurement induced squeezing can be
turned intoreal squeezing. This is discussed in detail in Ref.
[64].
3.5 Parametric instability
We have seen that for a driving laser red-detuned from thecavity
frequency c the upper sideband is resonantly en-hanced by the
cavity, which leads to preferred extractionof mechanical energy,
i.e. cavity cooling. For blue-detunedlight, in contrast, it is the
lower sideband that is reso-nantly enhanced by the cavity,
resulting in the preferreddeposition of mechanical energy, i.e. the
optical amplifi-cation of mechanical motion. Invoking the
rotating-wave
approximation for +m one finds that this processis described at
the simplest level by the 2-mode squeez-ing interaction (42)
instead of the beam-splitter Hamilto-nian (41) of section 3.2.
In that regime the optomechanical system can displaydynamical
instabilities. For appropriate parameters theyresult in stable
mechanical oscillations somewhat remi-niscent of laser action, but
for a phononic field [65, 66],or even in unstable dynamics and
chaos [67]. That thiscan be the case is already apparent at the
classical levelfrom the fact that opt can become negative for blue
de-tuning, see Eq. (34). If the laser intensity is strong
enoughthat the total damping rate +opt is itself negative, thenany
amplitude oscillation will grow exponentially until itsaturates due
to the onset of nonlinear effects.
Following Ludwig et al. [69] we assume that the motionof the
cantilever is approximately sinusoidal,
x(t ) x+ A cos(m t ), (55)
with the average position x given by the radiation
pressureforce,
x = 1m2m
Frad =G
m2m|(t )|2 (56)
where (t )|2 is the intracavity light intensity and A is
theamplitude of oscillations of the mirror. Marquardt andcoworkers
[71] showed that with Eq. (55), Eq. (8) yields forthe intracavity
field
(t )= e i(t )nne
inm t , (57)
with
n =(max
2
) Jn(GA/m)inm/+ i (Gx)/+1/2
. (58)
Here (t) = (GA/m)sin(m t) and Jn are Bessel func-tions of the
first kind. The stability of the system can bedetermined simply
comparing the mechanical power Praddue to radiation pressure to the
dissipated power Pfric dueto friction. When their ratio increases
above unity the sys-tem starts to undergo self-induced oscillations
[69].
Figure 6 is an example of a stability diagram deter-mined from
such an analysis. It shows the ratio Prad/Pfricas a function of the
detuning and the square of the(dimensionless) mechanical energy A2.
Regions withPrad/Pfric > 1 are unstable, and the solid line
defines an at-tractor where there is an exact power balance between
am-plification and damping. In general the parameter space(,A) is
characterized by the presence of a number of suchattractors. For
relatively weak amplitudes A, nonlineareffects tend to stabilize
the oscillations of the cantilever,
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-
P. Meystre: A short walk through quantum optomechanics
Figure 6 (Color online) Attractor diagram obtained from
therequirement that the optical power fed into mechanical
oscil-lations is balanced by the power lost to friction. Adapted
fromFrom Ref. [69], with permission.
leading to "laser-like" oscillations [65,66], but for larger
os-cillations amplitudes the system can become chaotic [67].Quantum
mechanically fluctuations are strongly ampli-fied just below
threshold, so that the attractor is no longersharp [70].
3.6 Quadratic coupling
So far we have considered geometries where the optome-chanical
coupling is linear in the oscillator displacement.Other forms of
coupling can however be considered, mostinterestingly perhaps a
coupling quadratic in the displace-ment. This can be realized in
so-called membrane-in-the-middle geometries, as first demonstrated
by J. Harrisand his collaborators at Yale [72, 73], see also Refs.
[74, 75].As implied by its name, this geometry involves an
oscil-lating mechanical membrane placed inside a Fabry-Protwith
fixed end-mirrors.
An attractive feature of membrane-in-the-middle con-figurations
is the ability to realize relatively easily eitherlinear or
quadratic optomechanical couplings, dependingon the precise
equilibrium position of the membrane. Incase the membrane is
located at an extremum ofc (x), sothatG =c/x = 0, see Eq. (6), we
have to lowest order
c (x)c +1
2
2c
x2(59)
so that the optomechanical Hamiltonian becomes
H =c aa+M bb+ 12
2c
x2x2zpt(b+ b)2aa. (60)
In the rotating wave approximation this reduces to
H = c aa+M bb+x2zpf2c
x2
(bb+1/2
)aa
= c aa+M bb+g (2)0(bb+1/2
)aa (61)
where
g (2)0 x2zpf2c
x2. (62)
Quadratic coupling opens up the way to a number of in-teresting
possibilities, including the direct measurementof energy
eigenstates of the mechanical element, ratherthan the position
detection characteristic of linear cou-pling. J. Harris and
coworkers estimate that it may be pos-sible in the future to use
this scheme to observe quantumjumps of a mechanical system [73]. In
another theoreti-cal study, Nunnenkamp and coworkers [76]
consideredoptomechanical cooling and squeezing via quadratic
op-tomechanical coupling. They showed that for high tem-peratures
and weak coupling, the steady-state phononnumber distribution is
nonthermal, and demonstratedhow to achieve mechanical squeezing by
driving the cav-ity with two optical fields.
Another possibility offered by that geometry is to ob-serve the
quantum tunneling of an optomechanical sys-tem operating deep in
the quantum regime through aclassically forbidden potential
barrier. One proposed ap-proach [77] relies on adiabatically
raising a potential bar-rier, whose parameters can be widely tuned,
at the lo-cation of a mechanical element. For the right choice
ofparameters the optomechanical potential is a
double-wellpotential, and it is estimated that quantum tunneling
be-tween its wells can occur at rates several orders of magni-tude
larger than the decoherence rate of the mechanicalmembrane. Besides
tunneling, that scheme may also al-low for the study of the quantum
Zeno effect in a mechan-ical context and provide a comparatively
simple schemefor the preparation and characterization of
non-classicalmechanical states of interest for quantum metrology
andsensing.
3.7 Pulsed optomechanics
So far we have largely limited our discussion to situationswhere
the optomechanical coupling is either constant orslowly varying in
time. One notable exception was the
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Ann. Phys. (Berlin) 0, No. 0 (2012)
quantum state transfer protocol outlined in Section III.B,which
requires that the interaction g (t) be turned off atthe precise
time when the state transfer has been com-pleted. However there are
a number of situations wherepulsed interactions are desirable, as
already realized byBraginsky [8, 78] in his proposal for a
back-action evadingposition measurement scheme. In a recent paper,
Vannerand coworkers [79] proposed to use a pulsed interactionof
duration short compared to the period of the mechani-cal oscillator
to generate and fully reconstruct quantumstates of mechanical
motion: As a result of the interac-tion the phase of the pulsed
driving optical field becomescorrelated with the position of the
mechanical oscillator,while its intensity imparts it a momentum
boost. A timedomain homodyne detection scheme can then be used
tomeasure the phase of the field emerging from the cavity,thereby
providing a measurement of the mechanical posi-tion. This scheme
can also be used to achieve squeezingand state purification of the
mechanical resonator. It hasalso recently been proposed that pulsed
optomechanicscould be used, at least in principle, to surpass the
limitsof conventional sideband cooling by using an
optimizedsequence of driving optical pulses [34, 35].
In an intriguing potential application of pulsed op-tomechanics,
Pikovski and colleagues [80] considered ascheme to measure the
canonical commutator of a mas-sive mechanical oscillator and by
doing so to detect pos-sible commutator deformations due to quantum
grav-ity: there are speculations that the existence of a min-imum
length scale where space-time is assumed to bequantized, possibly
of the order of the Planck lengthLP = 1.61035m, could result in
such deformations. Inthis proposal a sequence of optomechanical
interactionswould be used to map the commutator of the
mechanicalresonator onto an optical pulse. Remarkably the
analysisof Ref. [80] suggests that as a result Planck-scale
physicsmight be observable in a relatively mundane quantumoptics
experiment.
4 Cold atoms
In a development complementary to the research onnanoscale
mechanical systems, recent quantum optome-chanics experiments have
also manipulated and con-trolled at the quantum level the
center-of-mass degreesof freedom of ultracold atomic ensembles
[8184]. In thefollowing we restrict our discussion to the case of a
neutralatomic sample cooled well below its recoil temperatureand
trapped inside a single-mode Fabry-Prot resonator.This could be for
example a nearly homogeneous and col-
lisionless Bose-Einstein condensate (BEC) at T 0 or asample
cooled near the vibrational ground state of one ora few wells of
the optical lattice formed by the optical field.Side mode
excitations of the condensate in the first case,and the vibrational
motion of thermal atoms in the sec-ond case, provide formal analogs
of one or several movingmirrors.
To see how this works we consider first a generic
modelconsisting of a BEC at T = 0 trapped inside a Fabry-Protcavity
of length L and mode frequency c . The atoms ofmass M are driven by
a pump laser of frequency L andwave number k. When L is far detuned
from the atomictransition frequency a the excited electronic state
of theatoms can be adiabatically eliminated and the atoms in-teract
dispersively with the cavity field. In the dipole androtating-wave
approximations, the Hamiltonian describ-ing the interaction between
the atoms and the opticalfield is
H =Hatom+Hfield, (63)
where
Hfield =c aa (64)
and
Hbec =dx(x)
[p2x2M
+U0 cos2(kx)aa](x). (65)
Here (x) is the bosonic Schrdinger field operator forthe atoms,
a is the photon annihilation operator as before,and the atoms
interact with the light field via the familiaroff-resonant
coupling
U0 = g 2R/(L a), (66)
where gR is the single-photon Rabi frequency. As alwaysH should
be complemented by contributions describingthe external driving of
the cavity field, dissipation andcollisions.
When the light field can be approximated as a planewave the
atomic field operator can likewise be expandedin terms of plane
waves as
(x)= (1/pL)qbke
i qx , (67)
where bq and bq are annihilation and creation opera-
tors for atomic bosons with the momentum k, satisfyingthe
bosonic commutation relations [bq , b
q ] = q,q and
[bq , bq ]= 0.Consider for simplicity the case of scalar
bosonic
atoms: In the absence of light field and atT = 0 the ground
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P. Meystre: A short walk through quantum optomechanics
state of the sample would be a condensate with zero
mo-mentum,
|0 = (b0)N |0, (68)
but as a result of virtual transitions the atoms can acquirea
recoil momentum 2`k, where ` = 0,1,2, . . . In thelimit of low
photon numbers it is sufficient to considerthe lowest diffraction
order, ` = 1 and the atomic fieldoperator can be conveniently
expressed in terms of a zero-momentum component and a sine
mode,
b00(x)+ b22(x) (69)where 0(x) is the condensate wave function
and 2(x)=p
2cos(2kx). For very weak optical fields the occupationof the
sine mode remains much smaller the the zero-momentum mode, so that
b0 '
pN and b2b2N . Sub-
stituting then Eq. (69) into the Hamiltonian (65) and in aframe
rotating at the pump laser frequency the Hamilto-nian H becomes
Hom,bec = 4recb2b2+aa[+ g2(b2+ b2)
], (70)
where
g2 = (U0/2)pN/2 (71)
is the effective atom-field coupling constant, rec =K 2/2M the
recoil frequency, and =c L+U0N/2 isan effective Stark-shifted
detuning .
The reduced Hamiltonian (70) describes the couplingof two
oscillators, the cavity mode a and the momentumside mode b2 via the
optomechanical coupling g2aa(b2+b2). This shows that the condensate
momentum sidemode behaves formally like a moving mirror driven
bythe radiation pressure of the intracavity field, see Eq. (28)for
comparison.
A similar analogy can be established when consider-ing a sample
of ultracold atoms tightly confined to anharmonic trap of frequency
z centered at some locationz0 along the resonator axis. The
position of atom i is thenzi = z0+zi , and the vacuum Rabi
frequency with whichit interacts with the field is
gR (zi )= gR sin(0+2kzi ), (72)
where 0 = kz0 so that Eq. (66) becomes
U0 = g (zi )2
L a. (73)
Summing over all atoms in the sample and expandingthen the far
off-resonant atom-field interaction to lowest
order in Kzi one finds for L =c [85]H (c +NU0 sin20)aa+z
ibi bi
+ U0 sin(20)aa[
ikzi
](74)
where N is the number of atoms and the operator bi de-scribes
the annihilation of a phonon from the center-of-mass motion of atom
i .
The second line of the Hamiltonian (74) describes
theoptomechanical coupling of the intracavity optical fieldto the
collective atomic variable
kizi = kNZcm (75)
which is nothing but a measure of the normal mode ofthe sample,
its center of mass Zcm =N1i zi . For smalldisplacements that mode
can be described as a harmonicoscillator of frequency z and mass NM
. In this picture,the atom-field system is therefore modeled by the
optome-chanical Hamiltonian
Hom,at =c aa+z bb+gN (b+ b)aa, (76)
where b and b are bosonic annihilation and creationoperators for
the center-of-mass mode of motion of theatomic ensemble, zzpf =
/2Nmz andgN =NU0(K zzpf)sin20 (77)
with scales aspN . Quantum optomechanics experiments
with non-degenerate ultracold atoms samples have so farbeen
carried out principally in the group of D. Stamper-Kurn at UC
Berkeley, while T. Esslinger and coworkers atETH Zrich have
concentrated on the use of Bose conden-sates [86]. In a
trailblazing experiment [85] Purdy et al po-sitioned a sample of
cold atoms with sub-wavelength ac-curacy in a Fabry-Prot cavity to
demonstrate the tuningfrom linear to quadratic optomechanical
coupling fromthe linear to the quadratic coupling regime. The
Berke-ley group also observed the measurement back-actionresulting
from the quantum fluctuations of the opticalfield by measuring the
cavity-light-induced heating of theatomic ensemble [81], the first
observation of quantumback-action on a macroscopic mechanical
resonator atthe standard quantum limit. More recent work [87]
de-tected the asymmetric coherent scattering of light by
acollective mode of motion of a trapped ultracold gas with0.5
phonons of average excitation, a result that comple-ments the work
of Safavi-Naeini et al. [45] on the asymmet-ric absorption of light
by a nanomechanical solid-stateresonator, see section 2.4.
14 Copyright line will be provided by the publisher
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Ann. Phys. (Berlin) 0, No. 0 (2012)
150
100
50
0
15
10
5
0
1
0
-130 -120 -110 -100 -90 13012011010090Frequency (kHz)
Phot
on s
pect
rum
(pho
tons
)A)
B)
0.5
1.5
0.40.6
1
2
46
10
Phon
on o
ccup
atio
n
0.4 0.6 1 2 4 6 10 20Cooperativity
Figure 7 (Color online) (a) Asymmetric optical scattering
fromquantum collective motion, with the measured Stokes side-bands
[left panels, (red) circles] and anti- Stokes sidebands[right
panels, (blue) circles] at various mean phonon
numbers,characterized by the so-called cooperatively coefficientC .
Fromtop to bottomC= 9.6; 1.9; 0.4. (b) Measured phonon occupationvs
cooperativity. From Ref. [87], with permission.
Turning now to quantum degenerate gases, Brennekeet al. [82]
studied the dynamics of a Bose condensate of87Rb atoms trapped
inside a high-finesse Fabry-Prot anddriven by a feeble optical
field. This experiment demon-strated the optomechanical coupling of
a collective den-sity excitation of the condensate, showing that it
behavesprecisely as a mechanical oscillator coupled to the
cavityfield, in quantitative agreement with a cavity
optomechan-ical model of Eq. (70). These authors also succeeded
inapproaching the strong coupling regime of cavity optome-chanics,
where a single excitation of the mechanical os-
cillator substantially influences the cavity field. In
subse-quent work, the Bose condensate was irradiated from theside
of the optical resonator, resulting in the demonstra-tion of a
second-order quantum phase transition wherethe condensed atoms
enter a self-organized super-solidphase, a process mathematically
described by the Dickemodel of an ensemble of two-state systems
coupled toa single-mode electromagnetic field. In contrast to
thesituation in the usual Dicke model, where the two statesof
interest are two atomic electronic levels coupled by adipole
optical transition, in the present case the relevantstates are two
different momentum states coupled to thecavity field mode [88,
89].
5 Outlook Functionalization and hybridsystems
The rapid progress witnessed by quantum optomechan-ics makes it
increasingly realistic to consider the use ofmechanical systems
operating in the quantum regimeto make precise and accurate
measurements of feebleforces and fields [90]. In many cases, these
measurementsamount to the detection of exceedingly small
displace-ments, and in that context the remarkable potential
forfunctionalization of opto and electromechanical devicesis
particularly attractive. Their motional degree(s) of free-dom can
be coupled to a broad range of other physicalsystems, including
photons via radiation pressure froma reflecting surface, spin(s)
via coupling to a magneticmaterial, electric charges via the
interaction with a con-ducting surface, etc. In that way, the
mechanical elementcan serve as a universal transducer or
intermediary thatenables the coupling between otherwise
incompatiblesystems. This potential for functionalization also
suggeststhat quantum optomechanical systems have the potentialto
play an important role in classical and quantum infor-mation
processing, where transduction between differentinformation
carrying physical systems is crucial.
Much potential for the functionalization of optome-chanical
devices is offered by interfacing them with a sin-gle quantum
object. This could be an atom or a molecule,but also an artificial
atom such as a nitrogen vacancycenter (NV center) in diamond [91],
a superconductingqubit [41, 92, 93] or a Bose-Einstein condensate
[94]. Sev-eral theoretical proposals [91, 94100] and more
recentlyexperimental realizations [101, 102] involving atomic
sys-tems have been reported. For example, a recent experi-ment
[103] realized a hybrid optomechanical system bycoupling ultracold
atoms trapped in an optical lattice to amicromechanical membrane,
their coupling being medi-
Copyright line will be provided by the publisher 15
-
P. Meystre: A short walk through quantum optomechanics
ated by the light field. Both the effect of the membrane mo-tion
on the atoms and the back-action of the atomic mo-tion on the
membrane were observed. Singh and cowork-ers [104] considered a
variation on that scheme where aBose condensate is trapped inside a
Fabry-Prot with amoveable end mirror driven by a feeble optical
field. Theyshowed that under conditions where the optical field
canbe adiabatically eliminated one can achieve high fidelityquantum
state transfer between a momentum side modeof the condensate, see
Eq. (69), and the oscillating end-mirror.
Artificial atoms such as NV centers are of much inter-est for
hybrid optomechanical systems [91] due to theattractive combination
of their optical and electronic spinproperties. Their ground state
is a spin triplet [105] thatcan be optically initialized,
manipulated and read-out bya combination of optical and microwave
fields, and theyare characterized by remarkably long
room-temperaturecoherence times for solid-state systems. As such,
theyoffer much promise for applications e.g. in quantum
in-formation processing and ultrasensitive magnetometry,where the
spin is used as an atomic-sized magnetic sen-sor [106108]. In this
context, a spin-oscillator systemof particular interest consists of
a magnetized cantilevercoupled to the electronic spin of the NV
center. A recentexperiment by Arcizet and colleagues demonstrated
thecoupling of a nanomechanical oscillator to such a defectin a
diamond nanocrystal attached to its extremity [109].
In two further recent demonstrations of the potentialof hybrid
optomechanical systems, a mechanical oscil-lator was used to
achieve the coherent quantum controlof the spin of a single NV
center [111], and the coherentevolution of the spin of an NV center
was coupled to themotion of a magnetized mechanical resonator to
senseits motion with a precision below 6 picometers [110].
Theauthors of that experiment comment that it may soonbecome
possible to detect the mechanical zero-point fluc-tuations of the
oscillator.
More speculatively perhaps, micromechanical oscil-lators in the
quantum regime offer a route toward newtests of quantum theory at
unprecedented sizes and massscales. For instance, spatial quantum
superpositions ofmassive objects could be used to probe various
theo-ries of decoherence and shed new light on the transi-tion from
quantum to classical behavior: In contrast tothe generally accepted
view that it is technical issuessuch as environmental decoherence
that rapidly destroysuch superpositions in massive objects and
establishthe transition from the quantum to the classical
world,some authors [112116] have proposed collapse mod-els that are
associated with more fundamental mecha-nisms and the appearance of
new physical principles.
Bouwmeester [117] has pioneered the idea that
quantumoptomechanics experiments may shed light on this issueand on
possible unconventional decoherence processes,and in recent work
Romero-Isart has analyzed the require-ments to test some of these
models and discussed thefeasibility of a quantum optomechanical
implementationusing levitating dielectric nanospheres [118,
119].
Acknowledgements. This work is supported by the US Na-tional
Science Foundation, by the DARPA ORCHID andQuASAR programs through
grants from AFOSR and ARO,and by the US Army Research Office. We
acknowledge en-lightening discussions with numerous colleagues, in
particularM. Aspelmeyer, L. Buchmann, A. Clerk, H. Jing, M. Lukin,
R.Kanamoto, K. Schwab, H. Seok, S. Singh, S. Steinke, M.
Ven-galattore, E. M. Wright and K. Zhang.
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