a r X i v :0712.1618v 1 [p h y s i c s .o p t i c s ] 10 D e c 2007
Cavity Opto-Mechanics
T.J.Kippenberg ∗
Max Planck Institut f¨u r Quantenoptik,Garching,Germany
and K.J.Vahala 2†
California Institute of Technology,Pasadena,USA
The coupling of mechanical and optical degrees of freedom via radiation pressure has been a sub-ject of early rearch in the context of gravitational wave detection.Recent experimental advances
have allowed studying for the first time the modifications of mechanical dynamics provided by radi-ation pressure.This paper reviews the conquences of back-action of light confined in whispering-gallery dielectric micro-cavities,and prents a unified treatment of its two manifestations:notably the parametric instability (parametric amplification)and radiation pressure back-action cooling.Parametric instability offers a novel ”photonic clock”which is driven purely by the pressure of light.In contrast,radiati
on pressure cooling can surpass existing cryogenic technologies and offers cooling to phonon occupancies below unity and provides a route towards cavity Quantum Optomechanics
I.
INTRODUCTION
Recent years have witnesd a ries of developments at the interction of two,previously distinct subjects.Optical (micro-)cavities[1]and micro (nano)mechanical resonators[2],each a subject in their own right with a rich scientific and technological history,have,in a n,be-come entangled experimentally by the underlying mech-anism of optical,radiation pressure forces.The forces and their related physics have been of major interest in the field of atomic physics[3,4,5,6]for over 5decades and the emerging opto-mechanical context has many par-allels with this field.
The opto-mechanical coupling between a moving mir-ror and the radiation pressure of light has first appeared in the context of interferometric gravitational wave ex-periments.Owing to the discrete nature of photons,the quantum fluctuations of the radiation pressure forces give ri to the so called standard quantum limit [7,8,9].In addition to this “quantum back-action ”effect,the pio-neering work of V.Braginsky[10]predicted that the radi-ation pressure of light,confined within an interferometer (or res
onator),gives ri to the effect of dynamic back-action (which is a classical effect,in contrast to the afore-mentioned quantum back-action)owing to the finite cav-ity decay time.The resulting phenomena,which are the parametric instability (and associated mechanical oscilla-tion)and opto-mechanical back-action cooling reprent two sides of the same underlying “dynamic back-action”mechanism .Later,theoretical work has propod to us-ing the radiation-pressure coupling to realize quantum non-demolition measurements of the light field[11]and as a means to create non-classical states of the light field[12]and the mechanical system[13].It is noted that the ef-fect of dynamic back-action is of rather general relevance
2
diation pressure,the dynamic manifestations of radia-tion pressure forces on micro-and nano-mechanical ob-jects have only recently become an experimental real-ity.Curiously,while the theory of dynamic back ac-tion was motivated by consideration of precision mea-surement in the context of gravitational wave detection using large interferometers[19],thefirst obrvation of this mechanism was reported in2005at a vastly differ-ent size scale in microtoroid cavities[20,21,22].The obrvations were focud on the radiation pressure in-duced parametric instability[19].Subquently,the re-ver mechanism[23](back-action cooling)has been ex-ploited to
cool cantilevers[24,25,26],microtoroids[27] and macroscopic mirror modes[28,29]as well as me-chanical nano-membranes[30].We note that this tech-nique is different than the earlier demonstrated radiation-pressure feedback cooling[28,31],which us electronic feedback analogous to“Stochastic Cooling”[32]of ions in storage rings and which can also provide very efficient cooling as demonstrated in recent experiments[26,33, 34].Indeed,rearch in this subject has experienced a remarkable acceleration over the past three years as re-archers in diverfields such as optical microcavities[1], micro and nano-mechanical resonators[2]and quantum optics pursue a common t of scientific goals t for-ward by a decade-old theoretical framework.Indeed, there exists a rich theoretical history that considers the implications of optical forces in this new context. Subjects ranging from entanglement[35,36];genera-tion of squeezed states of light[12];to measurements at or beyond the standard quantum limit[11,37,38]; and even to tests of quantum theory itlf are in play here[39].On the practical side,there are opportuni-ties to harness the forces for new metrology tools[34] and even for new functions on a miconductor chip (e.g.,oscillators[20,21],optical mixers[40],and tune-able opticalfilters and switches[41,42].It ems clear that a newfield of cavity optomechanics has emerged, and will soon evolve into cavity quantum optomechanics (cavity QOM)who goal is the obrvation and explo-ration of quantum phenomena of mechanical systems[43] as well as quantum phenomena involving both photons and mechanical systems.The re
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alization of dynamical, opto-mechanical coupling in which radiation forces me-diate the interaction,is a natural outcome of underly-ing improvements in the technologies of optical(micro) cavities and mechanical micro(nano-)resonators.Re-duction of loss(increasing optical and mechanical Q) and reductions in form factor(modal volume)have en-abled a regime of operation in which optical forces are dominant[20,21,25,27,29,29,30,33,34,42].This coupling also requires coexistence of both high-Q opti-cal and high-Q mechanical modes.Such coexistence has been achieved in the geometries illustrated in Figure1. It also ems likely that other optical microcavity ge-ometries such as photonic crystals[44]can exhibit the dynamic back-action effect provided that structures are modified to support high-Q mechanical modes.
To understand how the coupling of optical and me-chanical degrees of freedom occurs in any of the depicted geometries,one need only consider the schematic in the upper panel of Figure1.Here,a Fabry-Perot optical cavity is assumed to feature a mirror that also functions as a is harmonically suspended). Such a configuration can indeed be encountered in grav-itational wave lar interferometers(such as LIGO)and is also,in fact,a direct reprentation of the“cantilever mirror”embodiment in the lower panel within Figure1. In addition it is functionally equivalent to the ca of a microtoroid embodiment(also shown in the lower panel), where the toroid
itlf provides both the optical modes as well as mechanical breathing modes(e Figure1and discussion below).Returning to the upper panel,inci-dent optical power that is resonant with a cavity mode creates a large circulating power within the cavity.This circulating power exerts a force upon the“mass-spring”mirror,thereby causing it to move.Reciprocally,the mir-ror motion results in a new optical round trip condition, which modifies the detuning of the cavity resonance with respect to the incidentfield.This will cau the system to simply establish a new,static equilibrium condition. The nonlinear nature of the coupling in such a ca can manifest itlf as a hysteretic behavior and was obrved over two decades ago in the work by Walther et.al[17]. However,if the mechanical and optical Q’s are sufficiently high(as is further detailed in what follows,such that the mechanical oscillation period is comparable or exceeds the cavity photon lifetime)a new t of dynamical phe-nomena can emerge,related to mechanical amplification and cooling.
In this paper,we will give thefirst unified treatment of this subject.Although microtoroid optical microcavities will be ud as an illustrative platform,the treatment and phenomena are universal and pertain to any cavity opto-mechanical systems supporting high Q optical and mechanical modes.In what follows we begin with a the-oretical framework through which dynamic back-action can be understood.The obrvation of micromechani-cal oscillation will then be considered by reviewing bot
h old and new experimental results that illustrate the ba-sic phenomena[20,21,22].Although this mechanism has been referred to as the parametric instability,we show that it is more properly defined in the context of me-chanical amplification and regenerative oscillation.For this reason,we introduce and define a mechanical gain, its spectrum,and,correspondingly,a threshold for oscil-lation.Mechanical cooling is introduced as the rever mechanism to amplification.We will then review the experimental obrvation of cooling by dynamic back-action[27,45]and also the quantum limits of cooling us-ing back action[46,47].
Finally,we will attempt to discuss some of the possible future directions for this newfield of rearch.
3 II.THEORETICAL FRAMEWORK OF
DYNAMIC BACK-ACTION
A.Coupled Mode Equations
Dynamic back-action is a modification of mechanical
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dynamics caud by non-adiabatic respon of the optical
field to changes in the cavity size.It can be understood
through the coupled equations of motion for optical and
mechanical modes,which can be derived from a single
Hamiltonian[48].
da
2τ0+
11
dt2+
Ωm
dt
+Ω2m x=
F RP(t)
m eff
=
ζ
T rt
+
F L(t)
2τ
=1
2τex
and∆(x)=ω−ω0(x)
accounts for the detuning of the pump lar frequency ωwith respect to the cavity resonanceω0(x)(w
hich,as shown below,depends on the mechanical coordinate,x). The power coupling rate into outgoing modes is described by the rate1/τex,whereas the intrinsic cavity loss rate is given by1/τ0.In the subquent discussion,the photon decay rate is also udκ≡1/τ.
The cond equation describes the mechanical coordi-nate(x)accounting for the movable cavity mirror),which is assumed to be harmonically bound and undergoing harmonic oscillation at frequencyΩm with a power dissipation rateΓm=Ωm
c
|a|2
T rt ).Moreover the term
F L(t)denotes the random Langevin force,and obeys F L(t)F L(t′) =Γm k B T R m effδ(t−t′),where k B is the Boltzmann constant and T R is the temperature of the rervoir.The Langevin force ensures that thefluctua-tion dissipation theorem is satisfied,such that the total steady state energy in the(classical)mechanical mode E m(in the abnce of lar radiation)is given by E m= Ω2m eff|χ(Ω)F L(Ω)|2dΩ=k B T R,where the mechan-
ical susceptibilityχ(Ω)=m−1
eff
/(iΩΓm+Ω2m−Ω2)has been introduced.Of special interest in thefirst equation is the optical detuning∆(x)which provides coupling to the cond equation through the relation:
∆(x)=∆+
ω0
x) of the above t of equations,it becomes directly evident that the equilibrium position of the mechanical oscillator will depend upon the intra-cavity power.Since the lat-ter is again coupled to the mechanical displacement,this leads to a cubic equation for the mirror position
τex|s|2=
cm eff
x 4τ2 ∆+ω0x 2+1 (4)
For sufficiently high power,this leads to bi-stable be-havior(namely for a given detuning and input power the mechanical position can take on veral possible values).
B.Modifications due to Dynamic Back-action:
Method of Retardation Expansion
The circulating optical power will vary in respon to changes in the coordinate”x”.The delineation of this respon into adiabatic and non-adiabatic contributions provides a starting point to understand the origin of dy-namic back-action and its two manifestations.This de-lineation is possible by formally integrating the above equation for the circulatingfield,and treating the term x wihin it as a perturbation.Furthermore,if the time variation of x is assumed to be slow on the time scale of the optical cavity decay time(or equivalently ifκ≫Ωm) then an expansion into orders of retardation is possible. Keeping only terms up to dx
4
in mechanical oscillator equation)can be expresd as follows,
mysP cav =
P 0cav
+
P 0cav
8∆τ2
R x −τP 0
cav
8∆τ2
furthermoreR
dx
T rt
=τ4∆2τ2+1|s |2denotes
the power in the cavity for zero mechanical displace-ment.The circulating power can also be written as P 0cav =F /π·C ·|s |2,where C =τ/τex
T rt denotes the cavity Fines.The first two
terms in this ries provide the adiabatic respon of the optical power to changes in position of the mir-ror.Intuitively,they give the instantaneous variations in coupled power that result as the cavity is “tuned”by changes in “x ”.It is apparent that the x −dependent contribution to this adiabatic respon (when input to the mechanical-oscillator equation-of-motion through the forcing function term)provides an optical-contribution to the stiffness of the mass-spring system.This so-called “optical spring”effect (or ”light
induced rigidity”)has been obrved in the context of LIGO[18]and also in microcavities[52].The corresponding change in spring constant leads to a frequency shift,relative to the un-pumped mechanical oscillator eigenfrequency,as given by (where P =|s |2denotes the input power):
∆Ωm =κ≫Ωm F 28n 2
ω0
(4τ2∆2+1)会计发展前景
P (6)
The non-adiabatic contribution in equation5is propor-tional to the velocity of the mass-spring system.When input to the mechanical-oscillator equation,the coeffi-cient of this term is paired with the intrinsic mechanical damping term and leads to the following damping rate given by (for a whispering gallery mode cavity of radius R ):子法
广州碧桂园学校Γ
=
κ≫Ωm
−F 3
8n 3
ω0R
(4τ2∆2+1)2
P (7)
Conquently,the modified (effective)damping rate of the mechanical oscillator is given by:
Γeff =Γ+Γm自学瑜伽
(8)
In both equation 6and 7we have stresd the fact that the expressions are valid only in the weak retar-dation regime in which κ≫Ωm .The sign of Γ(and the corresponding direction of power flow)depends upon the relative detuning of the optical pump with respect to the cavity resonant frequency.In particular,a red-detuned pump (∆<0)results in a sign such that op-tical forces augment intrinsic mechanical damping,while
FIG.2:The two manifestations of dynamic back-action:blue-detuned and red-detuned pump wave (green)with respect to
optical mode line-shape (blue)provide mechanical amplifica-tion and cooling,respectively.Also shown in the lower panels are motional sidebands (Stokes and anti-Stokes fields)gen-erated by mirror vibration and subquent Doppler-shifts of the circulating pump field.The amplitudes of the motional sidebands are asymmetric owing to cavity enhancement of the Doppler scattering process.
a blue-detuned pump (∆>0)revers the sign so that damping is offt (negative damping or amplification).It is important to note that the cooling rate in the weak-retardation regime depends strongly (∝F 3)on the opti-cal fines,which has been experimentally verified as dis-cusd in ction 4.2.Note also that maximum cooling or amplification rate for given power occurs when the lar is detuned to the maximum slope of the cavity lorenzian;the two cas are illustrated in Figure 1.The mod-ifications have been first derived by Braginsky[10]more than 3decades ago and are termed dynamic back-action .Specifically,an optical probe ud to ascertain the po-sition of a mirror within an optical resonator,will have the side-effect of altering the dynamical properties of the mirror (viewed as a mass-spring system).
The direction of power-flow is also determined by the sign of the pump frequency detuning relative to cavity resonant frequency.Damping (red tuning of the pump)is accompanied by power flow from the mechanical mode to the optical mode.This flow results in cooling of the mechanical mode.Amplification (blue tuning of pump)is,not surprisingly,accompanied by net power flow from the optical mode to the mechanical mode.This ca has also been referred to as heating ,however,it is more ap-propriately referred to as amplification since the power flow in this direction performs work on the mechanical mode.The nature of power flow between the mechani-cal and optical components
of the system will be explored here in veral ways,however,one form of analysis makes contact with the thermodynamic analogy of cycles in a Clapeyron or Watt diagram (i.e.a pressure-volume di-
5
FIG.3:Work done during one cycle of mechanical oscilla-
tion can be understood using a PV diagram for the radiation
pressure applied to a piston-mirror versus the mode volume
displaced during the cycle.In this diagram the cycle follows
a contour that circumscribes an area in PV space and hence
work is performed during the cycle.The n in which the
contour is traverd(clockwi or counterclockwi)depends
upon whether the pump is blue or red detuned with respect
to the optical mode.Positive work(amplification)or negative
work(cooling)are performed by the photon gas on the piston
mirror in the corresponding cas.
agram).In the prent ca–assuming the mechanical
oscillation period to be comparable to or longer than the
cavity lifetime–such a diagram can be constructed to an-
alyze powerflow resulting from cycles of the coherent ra-
diation gas interacting with a movable piston-mirror[53]
(e Figure3).In particular,a plot of radiation pressure
exerted on the piston-mirror versus changes in optical
mode volume provides a coordinate space in which it is
possible to understand the origin and sign of work done
during one oscillation cycle of the piston mirror.Consid-
ering the oscillatory motion of the piston-mirror at some
本二
eigenfrequencyΩm,then becau pressure(proportional
to circulating optical power)and displaced volume(pro-
portional to x)have a quadrature relationship(through
the dynamic back-action term involving the velocity dx
6 in accordance with the above
model.
FIG.4:Dynamics in the weak retardation regime.Experi-mental displacement spectral density function
s for a mechani-cal mode with eigenfrequency40.6MHz measured using three, distinct pump powers for both blue and red pump detuning. The mode is thermally excited(green data)and its linewidth can be en to narrow under blue pump detuning(red data) on account of the prence of mechanical gain(not sufficient in the prent measurement to excite full,regenerative oscil-lations);and to broaden under red pump detuning on account of radiation pressure damping(blue data).
Figure
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4prents such lineshape data taken using a microtoroid resonator.The power spectral density of the photocurrent is measured for a mechanical mode at an eigenfrequency of40.6MHz;three spectra are shown,corresponding to room temperature intrinsic ligible pumping),mechanical amplification and cooling.In addition to measurements of amplifi-cation and damping as a function of pump power(for fixed detuning),the dependence of the quantities on pump detuning(with pump powerfixed)can also be measured[27].Furthermore,pulling of the mechanical eigenfrequency(caud by the radiation pressure modi-fication to mechanical stiffness)can also be studied[52].
A summary of such data measured using a microtoroid in the regime whereκ Ωm is provided in Figure5. Both the ca of red-(cooling)and blue-(amplification) pump detuning are shown.Furthermore,it can be en that pump power was sufficient to drive the mechani-cal system into regenerative oscillation over a portion of the blue detuning region(ction of plot where linewidth is nearly zero).For comparison,the theoretical predic-tion is shown as the solid curve in the plots.Concerning radiation-pressure-induced stiffness,it should be noted that for red-detuning,the frequency is pulled to lower frequencies(stiffness is reduced)while for blue-detuning the stiffness increas and the mechanical eigenfrequency shifts to higher values.This is in dramatic contrast to similar changes that will be discusd in the next ction. While in Figure5the absolute shift in the m
echanical eigenfrequency is small compared toΩm,it is interesting to note that it is possible for this shift to be large.Specif-ically,statically unstable behavior is possible if the total spring constant reaches a negative value.This has in-
deed been obrved experimentally in gram scale mirrors coupled to strong intra-cavityfields by the LIGO group
at MIT[29].
FIG.5:Upper panel shows mechanical linewidth(δm=Γef f/2π)and shift in mechanical frequency(lower panel) measured versus pump wave detuning in the regime where Ωm≪κ.For negative(positive)detuning cooling(amplifica-tion)occurs.The region between the dashed lines denotes the ont of the parametric oscillation,where gain compensates mechanical loss.Figure stems from reference[27].Solid curves are theoretical predictions bad on the sideband model(e ction2.2).
While the above approach provides a convenient way
to understand the origin of gain and damping and their relationship to non-adiabatic respon,a more general
understanding of the dynamical behavior requires an ex-tension of the formalism.Cas where the mechanical
frequency,itlf,varies rapidly on the time scale of the cavity lifetime cannot be described correctly using the
above model.From the viewpoint of the sideband pic-
ture mentioned above,the modified formalism must in-clude the regime in which the sideband spectral para-
tion from the pump can be comparable or larger than the cavity linewidth.
C.Sideband Formalism
It is important to realize that the derivation of the
last ction only applies to the ca where the condi-tionκ≫Ωm is satisfied.In contrast,a perturbative
expansion of the coupled mode equations(eqns.1and 2)gives an improved description that is also valid in the
regime where the mechanical frequency is comparable to or even exceeds the cavity decay rate(whereΩm≫κis the so called resolved sideband regime).The derivation that leads to the results[27]is outlined here.For most
experimental considerations(in particular for the ca of
