ARTICLES

Resolved-sideband cooling of amicromechanical oscillatorA SCHLIESSER R RIVIERE G ANETSBERGER O ARCIZET AND T J KIPPENBERGMax-Planck-Institut fur Quantenoptik Hans-Kopfermann-Straszlige 1 85748 Garching Germanye-mail tjkmpqmpgde

Published online 13 April 2008 doi101038nphys939

In atomic laser cooling preparation of the motional quantum ground state has been achieved using resolved-sideband cooling oftrapped ions Here we report the first demonstration of resolved-sideband cooling of a mesoscopic mechanical oscillator a keystep towards ground-state cooling as quantum back-action is sufficiently suppressed in this scheme A laser drives the first lowersideband of an optical microcavity resonance the decay rate of which is twenty times smaller than the eigenfrequency of the associatedmechanical oscillator Cooling rates above 15 MHz are attained three orders of magnitude higher than the intrinsic dissipation rateof the mechanical device that is independently monitored at the 10minus18 m

radicHz level Direct spectroscopy of the motional sidebands

of the cooling laser confirms the expected suppression of motional increasing processes during cooling Moreover using two-modepumping this regime could enable motion measurement beyond the standard quantum limit and the concomitant generation of non-classical states of motion

In atomic laser cooling the lowest temperature that can be attainedfor a trapped ion (or atom) with an optical decay rate κ isgiven by TD

sim= hκ4kB the Doppler temperature1 If the harmonictrapping frequency Ωm is smaller than κ the minimum averageoccupation number 〈n〉 in the harmonic trapping potential is〈n〉 asymp κ4Ωm gt 1 that is the ionrsquos harmonic motion cannot becooled to the quantum ground state1 This fundamental limit canbe viewed as a direct consequence of the Heisenberg uncertaintyrelation2 a spontaneous emission which occurs over a timescale1κ implies an energy uncertainty of 1E sim hκ As the averageenergy of the oscillator 〈E〉 = hΩm(〈n〉+12) cannot be lowerthan its uncertainty this implies that the ground state is notreached when κ Ωm In a different perspective this fundamentaltemperature limit can also be interpreted as the quantum back-action3 that light exerts on the ion Owing to the discrete andstochastic nature of photons scattering events occur randomly intime and lead to a fluctuation of the radiation-pressure force thatentails heating of the ion and prevents ground-state cooling

Ground-state cooling is however possible in the resolved-sideband regime Resolved-sideband cooling45 requires aharmonically trapped dipole such as an atom or ion exhibitinga trapping frequency Ωm much larger that the optical resonancelinewidth κ thereby satisfying the so-called lsquostrong bindingconditionrsquo1 The physics of resolved-sideband cooling can beunderstood in a simple manner Owing to its harmonic motiona spatially oscillating excited ion will emit phase-modulatedradiation Consequently the emission spectrum consists of a seriesof sidebands at frequencies ω0 minus jΩm where j =plusmn1plusmn2 and ω0

is the unperturbed transition frequency Inversely the absorptionspectrum as probed by an observer in the laboratory frame willconsist of a series of absorption lines broadened owing to theupper statersquos decay rate (Fig 1a) Cooling can be achieved bytuning the incident laser radiation to one of the energeticallylower-lying sidebands for example ωL = ω0 minus Ωm As the ionabsorbs a photon of energy hωL = h(ω0 minusΩm) whereas it emitson average a photon of energy hω0 (neglecting recoil1) this entails

a reduction of its translational energy by one quantum leading tocooling The lowest occupancy that can be attained is given by6

〈n〉 asymp κ216Ω 2m 1 implying that the particle can be found in the

ground state most of the time Hence by making the energy scalehκ set by the spontaneous emission small in comparison with thelevel spacing of the trapped ion or atom hΩm ground-state coolingcan be achieved This powerful cooling technique first proposed5

in 1975 has been called lsquocooling by motional sideband excitationrsquo orlsquosideband coolingrsquo and fifteen years later has led to the remarkabledemonstration of ground-state cooling of trapped ions78

Importantly many of these considerations in atomic lasercooling also apply to electromechanics and the emerging field ofcavity optomechanics9 which study the dynamics of a mechanicaloscillator parametrically coupled to an electrically or opticallyresonant device Owing to the dynamical back-action310 of theelectrical or optical field on the mechanical oscillator in the formof a Coulomb or radiation-pressure force it is possible to cool themechanical mode when the resonant system is excited in a detunedmanner As in atomic laser cooling this effect is classical in natureas opposed to the quantum back-action previously discussedElemental back-action cooling was first observed experimentallyin 2006 for various micromechanical oscillators constituting theboundary of very high-finesse optical cavities11ndash13 thus enablingradiation-pressure cooling A similar phenomenon based onthermal effects has been reported earlier14 Back-action coolingwas also reported for a nanomechanical oscillator acting as thegate electrode of a superconducting single-electron transistor15Subsequently back-action cooling in optical experiments at a largerscale16 involving very-high-Q mechanical nanomembranes17 andelectromechanical cooling of a cantilever forming one plate ofa capacitor of a resonant electrical circuit18 were demonstratedOther anticipated embodiments such as mechanical oscillatorscoupled to an optical quantum dot19 trapped ions2021 a microwaveresonator2223 a superconducting quantum interference device24 ora resonant electronic circuit with an embedded Cooper pair box25

are predicted to exhibit similar phenomena

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 415

copy 2008 Nature Publishing Group

ARTICLES

ω L

ω L

ω0

ω L

ω ω0 ω

ω L+Ωm

ω Lndash Ωm

|0nndash1rang

|0n+1rang|0nrang

|1nndash1rang

|1n+1rang|1nrang

Ω lt m

Ωh mndash

ωh 0ndash

Ωnmin asymp 4 m gtgt 1 Ωnmin asymp 216 m2 lt 1

Ω gtgt m

a

b

κ

κ

κ κ

κ

Figure 1 Cooling a mechanical oscillator a Cooling proceeds by exciting theresonant (red curve) system coupled to the oscillator with a detuned source forexample a laser (blue line) When the resonant systemrsquos decay rate κ exceeds themechanical oscillatorrsquos resonance frequency Ωm ground-state cooling is impossible(left panel) owing to the quantum back-action If however Ωm κ the motionalsidebands are resolved and cooling can proceed to occupation far below unity (rightpanel) b The underlying transitions showing the ladder of the motional Fock statesof the mechanical oscillator coupled to the resonant systemrsquos transitions |0〉 rarr |1〉(ref 19) Cooling proceeds by predominant absorption on the red sideband (bluearrows) giving rise to a blue-shifted photon carrying away the energy from anannihilated phonon The opposite process (red arrows) leads to heating and issufficiently well suppressed only in the resolved-sideband regime

Interestingly however for virtually all studied systemstheoretical work15181925ndash28 has identified a cooling limit equivalentto the Doppler temperature in atomic physics caused by therandom momentum kicks of the discrete particles (photonselectrons or charged quasiparticles) involved in the coolingmechanism In analogy to atomic physics dynamic back-actioncooling of the oscillator to its motional ground state is impossibleif the mechanical frequency Ωm does not exceed the decay rateκ of the electromagnetic resonance Remarkably in spite ofthe prevalent awareness of the Doppler limit no electro- oroptomechanical experiment so far has been able to enter theresolved-sideband regime

Here we present resolved-sideband cooling of a mechanicaldevice the frequency Ωm of which exceeds the decay rate κ ofthe optical resonator it is coupled to by more than twenty timesThe system therefore operates in the resolved-sideband regimealso termed the lsquogood-cavity limitrsquo Similar to the ionrsquos case theparametric modulation of the resonance condition of the opticalcavity through the harmonic motion of its boundary gives rise tooptical sidebands in the cavity absorption spectrum at frequenciesω0 minus jΩm where j = plusmn1plusmn2 and ω0 is the unperturbed cavityresonance frequency The relative weights of the sidebands aredetermined by Bessel functions |Jj(β)|2 with β = ω0xΩmR in theoptomechanical case R being the radius of the device We notethat in the optomechanical case usually β 1 making it sufficientto consider only first-order sidebands In an energy picture thesidebands correspond to transitions in which the state of the cavityis changed (a photon is added) simultaneously with a change inthe occupation of the mechanical oscillator (Fig 1b) Motionaldecreasing transitions imply a reduction of phonon occupationn rarr n minus 1 when the photon is scattered into the cavity As onaverage the escape of the photon from the cavity does not changethe motional state the photon carries away the gained energy from

10 μm

Detuning (MHz)

= 094

= 147

= 175β

β

β

10 μm1 μm

1 μm

Optical WGM ( 0 )ω

Mechanical RBM ( m m)Ω Γ

ndash200 ndash100 0 +100 +200

Tran

smis

sion

(au

)

a

b c

κ

Figure 2 Resolved-sideband regime of a mesoscopic optomechanicaloscillator a Scanning electron microscope image of the system used consisting ofa silica microtoroidal optical cavity supporting both ultrahigh-finesse opticalresonances and high-Q radial breathing modes (RBMs) (Q= 30000) held by alsquoneedlersquo pillar An image of an intentionally broken cavity structure is also shownrevealing the ultrathin silicon support pillar with a diameter of 500 nm whichreduces the coupling to the pillar and thus enables high mechanical Q factorsb Schematic diagram of the radiation-pressure coupling between the optical andmechanical modes in the toroidal microcavity c Cavity transmission spectrum of amicrotoroid when its mechanical degree of freedom is excited with a coherent drive(using an auxiliary laser beam see the Supplementary Information) of differentamplitudes The optical cavity decay rate corresponds to κ2π= 32MHz whereasthe mechanical breathing mode exhibits a frequency of Ωm2π= 735MHzthereby placing the system deeply into the resolved-sideband regime The weightsof the sidebands follow the Bessel function expansion (solid line is a fit)

the mechanical mode leading to its cooling This process competeswith the reverse process where the phonon occupation is increasedn rarr n +1 by off-resonant absorption on the fist upper sidebandRigorous quantum mechanical calculations2728 show that motionalincreasing and decreasing processes occur with rates proportionalto (n + 1) middot A+ and n middot Aminus respectively where n is the phononoccupation number and Aplusmn

prop ((κ2)2+ (∆∓Ωm)2)minus1 with the

laser detuning ∆ = ωL minus ω0 from the cavity resonance Evidentlyif the laser is tuned to the lower sideband ∆ = minusΩm coolingis resonantly enhanced In the case of well-resolved sidebandsκ Ωm the motional decreasing transitions remain dominantalso for very small n as required for ground-state cooling andthe minimum achievable phonon number reproduces the result〈n〉 asymp κ216Ω 2

m 1 known from atomic physicsSilica whispering-gallery mode (WGM) resonators such

as toroidal microcavities29 have previously been shownto exhibit strong optomechanical coupling and dynamicalback-action effects30ndash33 By optimizing microfabrication we obtaintoroidal structures operating in the lsquostrong bindingrsquo conditionaccommodating high-quality-factor optical and mechanicalmodes in the same device (Fig 2b) The measured parametersΩm2π = 735 MHz and κ2π = 32 MHz corresponding to

416 nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics

copy 2008 Nature Publishing Group

ARTICLES

a

b

c

EOM

Spectrum analyser

Frequency lock

Vacuumchamber

NdYAG monitoring

laser

Coolingdiode laser

Shot-noise-limiteddisplacement sensor

Far-detuned cooling laser lock

WDMWDMFPC

FPC

Displacement calibration

IF

PBS

Detuning control

1064 nm

980 nm-toroid

Coolinglaser

Optical frequency

EcavToroidFibre

1064 nm

Monitoringlaser Monitoring

cavitymode

Coolingcavitymode

A+ Andash

λ 4λ 2

λ 2

μ

ϕ

ELO

Δ = 0 Δ Ω = ndash m

ΩmΩm

κ

Figure 3 Schematic diagram of the experiment a The optomechanical system(toroidal microcavity) is held in an evacuated chamber and is simultaneously excitedby a 980 nm diode laser that serves for cooling and a 1064 nm NdYAG laser forshot-noise-limited monitoring of the displacement of the cavityrsquos mechanical modesb The monitoring laser is polarized such that only a small fraction of its powercouples into the (polarization non-degenerate) WGM The stronger orthogonal fieldcomponent serves as a local oscillator in a polarization-sensitive detection schemeenabling shot-noise-limited displacement monitoring The absolute calibration of themeasured displacements is derived from a phase modulation of the monitoring laserwith a known modulation depth (see the Supplementary Information) c By meansof a frequency modulation technique the cooling laser is locked far outside thecavityrsquos WGM resonance to a detuning ∆= minusΩm such that anti-Stokes scatteringof photons into the cavity mode is resonantly enhanced The independent monitoringlaser is locked to the centre of a different resonance using the HanschndashCouillaudsignal (see the Supplementary Information) FPC fibre polarization controller WDMwavelength division multiplexer IF interference filter PBS polarizing beam splitterl2 l4 optical retarder plates

a finesse of F = 44 times 105 place the system deeply into theresolved-sideband regime The repercussions of this regime on thecavity transmission are most strikingly observed when an auxiliarylaser is used to drive the mechanical motion (using blue-detunedlight31 see the Supplementary Information) Indeed when tuningover the driven cavity a series of optical resonances spaced bythe mechanical frequency can be observed satisfying Ωmκ asymp 22(Fig 2c) which convincingly proves that the device satisfies thelsquostrong bindingrsquo condition

To implement resolved-sideband cooling a grating-stabilizedlaser diode is coupled to a high-finesse WGM near 970 nmof a second sample (Ωm2π = 406 MHz κ2π = 58 MHzΓm2π = 13 kHz meff = 10 ng) and locked to the first lowermotional sideband (∆ = minusΩm) using a frequency modulationtechnique and fast feedback (see Supplementary Informationand Fig 3a) The cooling caused by this laser is independentlyand continuously monitored by a NdYAG laser (l = 1064 nm)coupled to a different cavity mode An adaptation of the

HanschndashCouillaud polarization spectroscopy technique34 to thepresent experiment (see the Supplementary Information) enableslocking the monitoring laser to the centre of the resonance whichin conjunction with low power levels ensures that the readoutlaserrsquos effect on the mechanical oscillator motion is negligibleSimultaneously it provides a shot-noise-limited signal whichenables monitoring the displacement noise of the cavity caused bythe thermal excitation of the toroidrsquos different mechanical modes(Fig 4ab) The displacement sensitivity of this measurementreaches 10minus18 m

radicHz and is among the best values achieved so

far35 It would in principle enable observing the radiation-pressurequantum back-action once the sample is cooled to sufficiently lowtemperature (see the Supplementary Information) and eventuallyapproaching the standard quantum limit

Turning on the cooling laser (while keeping the readout laserunchanged) a clear reduction of the displacement fluctuations isobserved characteristic of cooling Note that our previous workhas already demonstrated unambiguously that cooling in toroidalmicrocavities is solely due to radiation pressure13 and thermalcontributions14 play negligible role We thus observe for the firsttime resolved-sideband cooling of a micromechanical oscillatorSimultaneous measurements on other radially symmetric modesat lower frequencies reveal that these remain unaffected (Fig 4c)This selectivity to a single mechanical mode is specific to theregime of resolved-sideband cooling In contrast to the lsquoweakbindingrsquo case where the κ-wide absorption sidebands of differentmechanical modes overlap resolved-sideband cooling can providehighly targeted cooling of only one mechanical mode As shownin Fig 4d the highest attained cooling rate Γc2π= 156 MHz wasachieved in the first sample (Ωm2π=735 MHz κ2π=32 MHz)for less than 3 mW of input power Note that only a fractionsim(4(Ωmκ)2

+1)minus1asymp 5 times 10minus4 of the launched power in the

fibre is coupled into the cavity (15 microW) owing to the highlydetuned working point Thus efficient cooling is achieved withextremely low powers coupled to the cavity Combining suchhigh cooling rates with the lowest achieved reservoir heatingrates of Γm2π = 13 kHz it seems feasible to attain a ratio ofinitial to final occupancy nRnf

sim= (Γc +Γm)Γm exceeding 103With the demonstrated 16Ω 2

mκ2= 7700 this value would be

sufficient to reach nf lt 05 when starting at a cryogenic lHetemperature of 18 K while still satisfying2728 nRnf lt Q equivΩmΓm

and Γc lt κ Starting from room temperature as reportedhere this would lead to nf lt 100 However analysis of theintegrated calibrated displacement noise spectra through therelation nf hΩm =

intmeffΩ 2Sx(Ω ) dΩ indicates significantly higher

nf This discrepancy is attributed to heating by excess phase noiseon the cooling laser beam Using an independent measurementit was measured to be as high as

radicSϕ asymp 4 microrad Hzminus12 even at

radiofrequencies close to Ωm (see the Supplementary Information)The resulting classical radiation-pressure back-action limits theachievable occupancy to nmin

sim=radic

2kBTmeffΓmSϕRΩmhω0 asymp 5200for the parameters of the second sample with which the lowestoccupancies were achieved Note that in this case the (classical)correlations between the laser noise and the induced displacementfluctuations can cause lsquosquashingrsquo36 artefacts if the diode laserwere also used for mechanical readout In contrast the use ofthe independent NdYAG laser provides a faithful displacementmonitoring with which these induced fluctuations can be revealedSuch an analysis yields a final occupancy of nf = 5900 in goodagreement with the above estimate In future experiments thistechnical limitation can be overcome by using quantum-noise-limited tunable solid-state lasers such as a Tisapphire laser

A direct consequence of the resolved-sideband regime is thestrong suppression of motional increasing processes which shouldlead to a suppression of the red sideband in the spectrum of the light

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 417

copy 2008 Nature Publishing Group

ARTICLES

Frequency (MHz)

0 1

0ndash50

+50

Time (μs)

x (fm

)

times 025

Frequency (MHz)1495 1500 2855 2860

δx (1

0ndash18

m H

zndash12

)δx

(10ndash1

8 m

Hzndash1

2)

40604055

δx (1

0ndash18

m H

zndash1

2 )

31 2

3

12

3

73 74 73 74

Frequency (MHz)Frequency (MHz)

Γ eff2π = 16 MHz

Pcool = 0 mW

1

10

100

1000

0 10 20 30 40

100

200

300

0

10

100

δx (1

0ndash18

m H

zndash12

)

10

100Pcool = 27 mW

a

b

c

d

Figure 4 Resolved-sideband cooling of the radial breathing modea Time-domain trace of the brownian motion of the radial breathing mode asobserved by the monitoring laser in a 2 MHz spectral bandwidth aroundΩm2π= 406MHz For observation times that are short compared with thecoherence time of the mechanical oscillator a sinusoidal oscillation is observedb Full spectrum of the displacement fluctuation spectral density δx at roomtemperature recorded with the NdYAG laser (red) The different peaks appearing inthe spectrum represent the mechanical eigenmodes that can be identified usingthree-dimensional finite-element analysis The modes denoted by (123) arerotationally symmetric mechanical modes the strain (colour code) and deformedshapes of which are shown in the insets The background of the measurement (grey)is due to shot noise its frequency dependence results from the reduceddisplacement sensitivity (for the same measured noise level) at frequenciesexceeding the cavityrsquos bandwidth A signal-to-background ratio close to 60 dB in thenoise power spectral density is achieved at room temperature c Resolved-sidebandcooling with the cooling laser tuned to the lower sideband of the radially symmetricradial breathing mode (3) As evident only mode (3) is cooled whereas all othermodes (of which 1 and 2 are shown) remain unaffected Circles represent noisespectra with the cooling laser off (red) and running at 300 microW (blue) Lines arelorentzian fits d Cooling rates exceeding 15 MHz obtained with the 735 MHz radialbreathing mode of a different sample

emerging from the cavity (as analysed theoretically in ref 27) Toconfirm this aspect the motional sidebands generated during thecooling cycles were probed similar to spectroscopy of the resonancefluorescence of a cooled ion37 This is achieved with a heterodyne

Cool

ing

lase

r

BS

BS

AOM200 MHz

Heterodyne detection

Frequency (MHz)

15935 2406024055 240651594515940

PSD

(dB)

ndash10

ndash20

ω Lndash Ω AOMω L

ω L

ω L

plusmn ΩmΩ AOM

Ωm Ωm

a

b

-toroidμ

Figure 5 Motional sideband spectroscopy a Experimental set-up used to resolveupper and lower motional sidebands generated during interaction of the coolinglaser with the cavity similar to the spectroscopy of the resonance fluorescence of alaser-cooled ion37 The cooling laser interacts with the optical microcavity thetransmission of which is subsequently superimposed with a second laser beamretrieved from the same cooling laser but down-shifted by 200MHz using anacousto-optic modulator The beating of the two signals is recorded using a balancedheterodyne detector yielding spectral components of the lower and upper sidebandsat 200MHzminusΩm2π and 200MHz+Ωm2π respectively BS beam splitterb Beat signals of the upper (anti-Stokes) and lower (Stokes) motional sidebands for∆= 0 (red) and a detuning close to ∆sim= minusΩm (blue) The plotted electrical noisepower spectral density (PSD) is proportional to the optical power spectral density inthe sidebands For zero detuning of the pump with respect to the optical cavity themotional sidebands are equal in power By tuning the laser to the lower sidebandwhich induced cooling strong reduction of the Stokes sideband is observed

experiment by beating the cooling laser with a local oscillator(Fig 5a) derived by down-shifting part of the cooling laser lightusing an acousto-optic modulator at ΩAOM2π = 200 MHz Thebeat of the local oscillator and the cooling laser produces amodulation at ΩAOM whereas the motional sidebandsrsquo signals nowappear at ΩAOM plusmnΩm thereby enabling measuring their individualweights Figure 5b shows the result of this measurement for twodifferent laser detunings Whereas for excitation on the cavity linecentre (∆=0) the sideband intensities are equal (Aminusn asymp A+(n+1)because n 1 and Aminus

= A+) detuning the laser to the lowersideband ∆=minusΩm should lead to a strong suppression of the red-sideband beat by a factor of AminusA+ sim= 16Ω 2

mκ2 In the experimentwith the 406 MHz sample the detuning is chosen such thatthe red sideband is still discernible above the laser phase noisecorresponding to a suppression of more than 15 dB Optimizingthe laser detuning the red emission sideband could be reducedeven further It is important to note that the ability to measurethe individual sidebands separatelymdashas demonstrated heremdashisimportant for future experiments that venture into the quantumregime As theoretically predicted27mdashand in analogy to trappedions738mdashthe weights of the sidebands enable inferring the averagemotional occupation number27 for low occupancies by measuringthe ratio between the red and blue sidebands

The regime of resolved sidebands has another importantmdashand counterintuitivemdashbenefit because the cooling rate is indeedhigher in comparison with the unresolved case Keeping the

418 nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics

copy 2008 Nature Publishing Group

ARTICLES

launched power P as well as Ωm and R fixed an increase inthe cavity finesse increases the cooling rate until it saturates inthe highly resolved-sideband case and approaches an asymptoticvalue (see the Supplementary Information) The circulating powerhowever continues to decrease mitigating undesired effects suchas photothermal14- or radiation-pressure-induced bistability39 orabsorption-induced heating It is important to note however thatthe final occupancies for very high optical finesse are bound by thecavity decay rate28

Pertaining to the wider implications of our work we notethat resolved-sideband cooling as demonstrated here is a keyprerequisite for ground-state cooling and is an enabling stepfor both electromechanical and optomechanical schemes towardsobserving quantum mechanical phenomena of macroscopicobjects4041 such as the generation of non-classical states ofmotion Equally important the regime of resolved sidebandsenables a measurement scheme that permits measuring only onemotion quadrature To this end a lsquocontinuous two-transducermeasurementrsquo3 is required using a simultaneous excitation at boththe first upper and lower motional sidebands Theoretically thisscheme enables exceeding the standard quantum limit by

radicΩmκ

a factor of about 5 for the parameters reported here Alongthese lines continuous quantum non-demolition measurementsof a mechanical oscillator may produce squeezed states ofmechanical motion42

Received 20 January 2008 accepted 7 March 2008 published 13 April 2008

References1 Wineland D J amp Itano W M Laser cooling of atoms Phys Rev A 20 1521ndash1540 (1979)2 Stenholm S The semiclassical theory of laser cooling Rev Mod Phys 58 699ndash739 (1986)3 Braginsky V B amp Khalili F Y Quantum Measurement (Cambridge Univ Press Cambridge 1992)4 Dehmelt H G Entropy reduction by motional sideband excitation Nature 262 777 (1976)5 Wineland D amp Dehmelt H Proposed 10141υ lt υ laser fluorescence spectroscopy on

T1+mono-ion oscillator III Bull Am Phys Soc 20 637ndash637 (1975)6 Neuhauser W Hohenstatt M Toschek P amp Dehmelt H Optical-sideband cooling of visible atom

cloud confined in parabolic well Phys Rev Lett 41 233ndash236 (1978)7 Diedrich F Bergquist J C Itano W M amp Wineland D J Laser cooling to the zero-point energy of

motion Phys Rev Lett 62 403ndash406 (1989)8 Monroe C et al Resolved-side-band Raman cooling of a bound atom to the 3d zero-point energy

Phys Rev Lett 75 4011ndash4014 (1995)9 Kippenberg T J amp Vahala K J Cavity opto-mechanics Opt Express 15 17172ndash17205 (2007)10 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)11 Arcizet O et al Radiation-pressure cooling and optomechanical instability of a micromirror Nature

444 71ndash74 (2006)12 Gigan S et al Self-cooling of a micromirror by radiation pressure Nature 444 67ndash70 (2006)13 Schliesser A et al Radiation pressure cooling of a micromechanical oscillator using dynamical

backaction Phys Rev Lett 97 243905 (2006)14 Metzger C H amp Karrai K Cavity cooling of a microlever Nature 432 1002ndash1005 (2004)15 Naik A et al Cooling a nanomechanical resonator with quantum back-action Nature 443

193ndash196 (2006)16 Corbitt T et al An all-optical trap for a gram-scale mirror Phys Rev Lett 98 150802 (2007)

17 Thomson J D et al Strong dispersive coupling of a high finesse cavity to a micromechanicalmembrane Nature 452 72ndash75 (2008)

18 Brown K R et al Passive cooling of a micromechanical oscillator with a resonant electric circuitPhys Rev Lett 99 137205 (2007)

19 Wilson-Rae I Zoller P amp Imamoglu A Laser cooling of a nanomechanical resonator mode to itsquantum ground state Phys Rev Lett 92 075507 (2004)

20 Wineland D et al Experimental issues in coherent quantum-state manipulation of trapped atomicions J Res Natl Inst Standards Technol 103 259ndash328 (1998)

21 Tian L amp Zoller P Coupled ion-nanomechanical systems Phys Rev Lett 93 266403 (2004)22 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)23 Regal C A Teufel J D amp Lehnert K W Measuring nanomechanical motion with a microwave

cavity interferometer Preprint at lthttparXiv08011827gt (2008)24 Blencowe M P amp Buks E Quantum analysis of a linear dc SQUID mechanical displacement

detector Phys Rev B 76 014511 (2007)25 Martin I Shnirman A Tian L amp Zoller P Ground-state cooling of mechanical resonators

Phys Rev B 69 125339 (2004)26 Blencowe M P Imbers J amp Armour A D Dynamics of a nanomechanical resonator coupled to a

superconducting single-electron transistor New J Phys 7 236 (2005)27 Wilson-Rae I Nooshi N Zwerger W amp Kippenberg T J Theory of ground state cooling of a

mechanical oscillator using dynamical backaction Phys Rev Lett 99 093902 (2007)28 Marquardt F Chen J P Clerk A A amp Girvin S M Quantum theory of cavity-assisted sideband

cooling of mechanical motion Phys Rev Lett 99 093902 (2007)29 Armani D K Kippenberg T J Spillane S M amp Vahala K J Ultra-high-Q toroid microcavity on a

chip Nature 421 925ndash928 (2003)30 Carmon T et al Temporal behavior of radiation-pressure-induced vibrations of an optical

microcavity phonon mode Phys Rev Lett 94 223902 (2005)31 Kippenberg T J et al Analysis of radiation-pressure induced mechanical oscillation of an optical

microcavity Phys Rev Lett 95 033901 (2005)32 Rokhsari H Kippenberg T J Carmon T amp Vahala K J Radiation-pressure-driven

micro-mechanical oscillator Opt Express 13 5293ndash5301 (2005)33 Ma R et al Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres

Opt Lett 32 2200ndash2202 (2007)34 Hansch T W amp Couillaud B Laser frequency stabilization by polarization spectroscopy of a

reflecting reference cavity Opt Commun 35 441ndash444 (1980)35 Arcizet O et al High-sensitivity optical monitoring of a micromechanical resonator with a

quantum-limited optomechanical sensor Phys Rev Lett 97 133601 (2006)36 Poggio M Degen C L Mamin H J amp Rugar D Feedback cooling of a cantileverrsquos fundamental

mode below 5 mK Phys Rev Lett 99 017201 (2007)37 Raab C et al Motional sidebands and direct measurement of the cooling rate in the resonance

fluorescence of a single trapped ion Phys Rev Lett 85 538ndash541 (2000)38 Leibfried D Blatt R Monroe C amp Wineland D Quantum dynamics of single trapped ions Rev

Mod Phys 75 281ndash324 (2003)39 Dorsel A McCullen J D Meystre P Vignes E amp Walther H Optical bistability and mirror

confinement induced by radiation pressure Phys Rev Lett 51 1550ndash1553 (1983)40 Schwab K C amp Roukes M L Putting mechanics into quantum mechanics Phys Today 58

36ndash42 (2005)41 Mancini S Giovannetti V Vitali D amp Tombesi P Entangling macroscopic oscillators exploiting

radiation pressure Phys Rev Lett 88 120401 (2002)42 Kuzmich A Mandel L amp Bigelow N P Generation of spin squeezing via continuous quantum

nondemolition measurement Phys Rev Lett 85 1594ndash1597 (2000)

Supplementary Information accompanies this paper on wwwnaturecomnaturephysics

AcknowledgementsThe authors acknowledge discussions with T W Hansch W Zwerger and I Wilson-Rae TJKacknowledges support through an Independent Max Planck Junior Research Group Grant a MarieCurie Excellence Grant (JRG-UHQ) the DFG-funded Nanosystems Initiative Munich (NIM) and aMarie Curie Reintegration Grant (RG-UHQ) The authors gratefully acknowledge J Kotthaus foraccess to clean-room facilities for microfabrication and A Marx for support with scanningelectron microscopy

Author informationReprints and permission information is available online at httpnpgnaturecomreprintsandpermissionsCorrespondence and requests for materials should be addressed to TJK

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 419

copy 2008 Nature Publishing Group

ARTICLES

a

b

c

EOM

Spectrum analyser

Frequency lock

Vacuumchamber

NdYAG monitoring

laser

Coolingdiode laser

Shot-noise-limiteddisplacement sensor

Far-detuned cooling laser lock

WDMWDMFPC

FPC

Displacement calibration

IF

PBS

Detuning control

1064 nm

980 nm-toroid

Coolinglaser

Optical frequency

EcavToroidFibre

1064 nm

Monitoringlaser Monitoring

cavitymode

Coolingcavitymode

A+ Andash

λ 4λ 2

λ 2

μ

ϕ

ELO

Δ = 0 Δ Ω = ndash m

ΩmΩm

κ

Figure 3 Schematic diagram of the experiment a The optomechanical system(toroidal microcavity) is held in an evacuated chamber and is simultaneously excitedby a 980 nm diode laser that serves for cooling and a 1064 nm NdYAG laser forshot-noise-limited monitoring of the displacement of the cavityrsquos mechanical modesb The monitoring laser is polarized such that only a small fraction of its powercouples into the (polarization non-degenerate) WGM The stronger orthogonal fieldcomponent serves as a local oscillator in a polarization-sensitive detection schemeenabling shot-noise-limited displacement monitoring The absolute calibration of themeasured displacements is derived from a phase modulation of the monitoring laserwith a known modulation depth (see the Supplementary Information) c By meansof a frequency modulation technique the cooling laser is locked far outside thecavityrsquos WGM resonance to a detuning ∆= minusΩm such that anti-Stokes scatteringof photons into the cavity mode is resonantly enhanced The independent monitoringlaser is locked to the centre of a different resonance using the HanschndashCouillaudsignal (see the Supplementary Information) FPC fibre polarization controller WDMwavelength division multiplexer IF interference filter PBS polarizing beam splitterl2 l4 optical retarder plates

a finesse of F = 44 times 105 place the system deeply into theresolved-sideband regime The repercussions of this regime on thecavity transmission are most strikingly observed when an auxiliarylaser is used to drive the mechanical motion (using blue-detunedlight31 see the Supplementary Information) Indeed when tuningover the driven cavity a series of optical resonances spaced bythe mechanical frequency can be observed satisfying Ωmκ asymp 22(Fig 2c) which convincingly proves that the device satisfies thelsquostrong bindingrsquo condition

To implement resolved-sideband cooling a grating-stabilizedlaser diode is coupled to a high-finesse WGM near 970 nmof a second sample (Ωm2π = 406 MHz κ2π = 58 MHzΓm2π = 13 kHz meff = 10 ng) and locked to the first lowermotional sideband (∆ = minusΩm) using a frequency modulationtechnique and fast feedback (see Supplementary Informationand Fig 3a) The cooling caused by this laser is independentlyand continuously monitored by a NdYAG laser (l = 1064 nm)coupled to a different cavity mode An adaptation of the

HanschndashCouillaud polarization spectroscopy technique34 to thepresent experiment (see the Supplementary Information) enableslocking the monitoring laser to the centre of the resonance whichin conjunction with low power levels ensures that the readoutlaserrsquos effect on the mechanical oscillator motion is negligibleSimultaneously it provides a shot-noise-limited signal whichenables monitoring the displacement noise of the cavity caused bythe thermal excitation of the toroidrsquos different mechanical modes(Fig 4ab) The displacement sensitivity of this measurementreaches 10minus18 m

radicHz and is among the best values achieved so

far35 It would in principle enable observing the radiation-pressurequantum back-action once the sample is cooled to sufficiently lowtemperature (see the Supplementary Information) and eventuallyapproaching the standard quantum limit

Turning on the cooling laser (while keeping the readout laserunchanged) a clear reduction of the displacement fluctuations isobserved characteristic of cooling Note that our previous workhas already demonstrated unambiguously that cooling in toroidalmicrocavities is solely due to radiation pressure13 and thermalcontributions14 play negligible role We thus observe for the firsttime resolved-sideband cooling of a micromechanical oscillatorSimultaneous measurements on other radially symmetric modesat lower frequencies reveal that these remain unaffected (Fig 4c)This selectivity to a single mechanical mode is specific to theregime of resolved-sideband cooling In contrast to the lsquoweakbindingrsquo case where the κ-wide absorption sidebands of differentmechanical modes overlap resolved-sideband cooling can providehighly targeted cooling of only one mechanical mode As shownin Fig 4d the highest attained cooling rate Γc2π= 156 MHz wasachieved in the first sample (Ωm2π=735 MHz κ2π=32 MHz)for less than 3 mW of input power Note that only a fractionsim(4(Ωmκ)2

+1)minus1asymp 5 times 10minus4 of the launched power in the

fibre is coupled into the cavity (15 microW) owing to the highlydetuned working point Thus efficient cooling is achieved withextremely low powers coupled to the cavity Combining suchhigh cooling rates with the lowest achieved reservoir heatingrates of Γm2π = 13 kHz it seems feasible to attain a ratio ofinitial to final occupancy nRnf

sim= (Γc +Γm)Γm exceeding 103With the demonstrated 16Ω 2

mκ2= 7700 this value would be

sufficient to reach nf lt 05 when starting at a cryogenic lHetemperature of 18 K while still satisfying2728 nRnf lt Q equivΩmΓm

and Γc lt κ Starting from room temperature as reportedhere this would lead to nf lt 100 However analysis of theintegrated calibrated displacement noise spectra through therelation nf hΩm =

intmeffΩ 2Sx(Ω ) dΩ indicates significantly higher

nf This discrepancy is attributed to heating by excess phase noiseon the cooling laser beam Using an independent measurementit was measured to be as high as

radicSϕ asymp 4 microrad Hzminus12 even at

radiofrequencies close to Ωm (see the Supplementary Information)The resulting classical radiation-pressure back-action limits theachievable occupancy to nmin

sim=radic

2kBTmeffΓmSϕRΩmhω0 asymp 5200for the parameters of the second sample with which the lowestoccupancies were achieved Note that in this case the (classical)correlations between the laser noise and the induced displacementfluctuations can cause lsquosquashingrsquo36 artefacts if the diode laserwere also used for mechanical readout In contrast the use ofthe independent NdYAG laser provides a faithful displacementmonitoring with which these induced fluctuations can be revealedSuch an analysis yields a final occupancy of nf = 5900 in goodagreement with the above estimate In future experiments thistechnical limitation can be overcome by using quantum-noise-limited tunable solid-state lasers such as a Tisapphire laser

A direct consequence of the resolved-sideband regime is thestrong suppression of motional increasing processes which shouldlead to a suppression of the red sideband in the spectrum of the light

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 417

copy 2008 Nature Publishing Group

ARTICLES

Frequency (MHz)

0 1

0ndash50

+50

Time (μs)

x (fm

)

times 025

Frequency (MHz)1495 1500 2855 2860

δx (1

0ndash18

m H

zndash12

)δx

(10ndash1

8 m

Hzndash1

2)

40604055

δx (1

0ndash18

m H

zndash1

2 )

31 2

3

12

3

73 74 73 74

Frequency (MHz)Frequency (MHz)

Γ eff2π = 16 MHz

Pcool = 0 mW

1

10

100

1000

0 10 20 30 40

100

200

300

0

10

100

δx (1

0ndash18

m H

zndash12

)

10

100Pcool = 27 mW

a

b

c

d

Figure 4 Resolved-sideband cooling of the radial breathing modea Time-domain trace of the brownian motion of the radial breathing mode asobserved by the monitoring laser in a 2 MHz spectral bandwidth aroundΩm2π= 406MHz For observation times that are short compared with thecoherence time of the mechanical oscillator a sinusoidal oscillation is observedb Full spectrum of the displacement fluctuation spectral density δx at roomtemperature recorded with the NdYAG laser (red) The different peaks appearing inthe spectrum represent the mechanical eigenmodes that can be identified usingthree-dimensional finite-element analysis The modes denoted by (123) arerotationally symmetric mechanical modes the strain (colour code) and deformedshapes of which are shown in the insets The background of the measurement (grey)is due to shot noise its frequency dependence results from the reduceddisplacement sensitivity (for the same measured noise level) at frequenciesexceeding the cavityrsquos bandwidth A signal-to-background ratio close to 60 dB in thenoise power spectral density is achieved at room temperature c Resolved-sidebandcooling with the cooling laser tuned to the lower sideband of the radially symmetricradial breathing mode (3) As evident only mode (3) is cooled whereas all othermodes (of which 1 and 2 are shown) remain unaffected Circles represent noisespectra with the cooling laser off (red) and running at 300 microW (blue) Lines arelorentzian fits d Cooling rates exceeding 15 MHz obtained with the 735 MHz radialbreathing mode of a different sample

emerging from the cavity (as analysed theoretically in ref 27) Toconfirm this aspect the motional sidebands generated during thecooling cycles were probed similar to spectroscopy of the resonancefluorescence of a cooled ion37 This is achieved with a heterodyne

Cool

ing

lase

r

BS

BS

AOM200 MHz

Heterodyne detection

Frequency (MHz)

15935 2406024055 240651594515940

PSD

(dB)

ndash10

ndash20

ω Lndash Ω AOMω L

ω L

ω L

plusmn ΩmΩ AOM

Ωm Ωm

a

b

-toroidμ

Figure 5 Motional sideband spectroscopy a Experimental set-up used to resolveupper and lower motional sidebands generated during interaction of the coolinglaser with the cavity similar to the spectroscopy of the resonance fluorescence of alaser-cooled ion37 The cooling laser interacts with the optical microcavity thetransmission of which is subsequently superimposed with a second laser beamretrieved from the same cooling laser but down-shifted by 200MHz using anacousto-optic modulator The beating of the two signals is recorded using a balancedheterodyne detector yielding spectral components of the lower and upper sidebandsat 200MHzminusΩm2π and 200MHz+Ωm2π respectively BS beam splitterb Beat signals of the upper (anti-Stokes) and lower (Stokes) motional sidebands for∆= 0 (red) and a detuning close to ∆sim= minusΩm (blue) The plotted electrical noisepower spectral density (PSD) is proportional to the optical power spectral density inthe sidebands For zero detuning of the pump with respect to the optical cavity themotional sidebands are equal in power By tuning the laser to the lower sidebandwhich induced cooling strong reduction of the Stokes sideband is observed

experiment by beating the cooling laser with a local oscillator(Fig 5a) derived by down-shifting part of the cooling laser lightusing an acousto-optic modulator at ΩAOM2π = 200 MHz Thebeat of the local oscillator and the cooling laser produces amodulation at ΩAOM whereas the motional sidebandsrsquo signals nowappear at ΩAOM plusmnΩm thereby enabling measuring their individualweights Figure 5b shows the result of this measurement for twodifferent laser detunings Whereas for excitation on the cavity linecentre (∆=0) the sideband intensities are equal (Aminusn asymp A+(n+1)because n 1 and Aminus

= A+) detuning the laser to the lowersideband ∆=minusΩm should lead to a strong suppression of the red-sideband beat by a factor of AminusA+ sim= 16Ω 2

mκ2 In the experimentwith the 406 MHz sample the detuning is chosen such thatthe red sideband is still discernible above the laser phase noisecorresponding to a suppression of more than 15 dB Optimizingthe laser detuning the red emission sideband could be reducedeven further It is important to note that the ability to measurethe individual sidebands separatelymdashas demonstrated heremdashisimportant for future experiments that venture into the quantumregime As theoretically predicted27mdashand in analogy to trappedions738mdashthe weights of the sidebands enable inferring the averagemotional occupation number27 for low occupancies by measuringthe ratio between the red and blue sidebands

The regime of resolved sidebands has another importantmdashand counterintuitivemdashbenefit because the cooling rate is indeedhigher in comparison with the unresolved case Keeping the

418 nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics

copy 2008 Nature Publishing Group

ARTICLES

launched power P as well as Ωm and R fixed an increase inthe cavity finesse increases the cooling rate until it saturates inthe highly resolved-sideband case and approaches an asymptoticvalue (see the Supplementary Information) The circulating powerhowever continues to decrease mitigating undesired effects suchas photothermal14- or radiation-pressure-induced bistability39 orabsorption-induced heating It is important to note however thatthe final occupancies for very high optical finesse are bound by thecavity decay rate28

Pertaining to the wider implications of our work we notethat resolved-sideband cooling as demonstrated here is a keyprerequisite for ground-state cooling and is an enabling stepfor both electromechanical and optomechanical schemes towardsobserving quantum mechanical phenomena of macroscopicobjects4041 such as the generation of non-classical states ofmotion Equally important the regime of resolved sidebandsenables a measurement scheme that permits measuring only onemotion quadrature To this end a lsquocontinuous two-transducermeasurementrsquo3 is required using a simultaneous excitation at boththe first upper and lower motional sidebands Theoretically thisscheme enables exceeding the standard quantum limit by

radicΩmκ

a factor of about 5 for the parameters reported here Alongthese lines continuous quantum non-demolition measurementsof a mechanical oscillator may produce squeezed states ofmechanical motion42

Received 20 January 2008 accepted 7 March 2008 published 13 April 2008

References1 Wineland D J amp Itano W M Laser cooling of atoms Phys Rev A 20 1521ndash1540 (1979)2 Stenholm S The semiclassical theory of laser cooling Rev Mod Phys 58 699ndash739 (1986)3 Braginsky V B amp Khalili F Y Quantum Measurement (Cambridge Univ Press Cambridge 1992)4 Dehmelt H G Entropy reduction by motional sideband excitation Nature 262 777 (1976)5 Wineland D amp Dehmelt H Proposed 10141υ lt υ laser fluorescence spectroscopy on

T1+mono-ion oscillator III Bull Am Phys Soc 20 637ndash637 (1975)6 Neuhauser W Hohenstatt M Toschek P amp Dehmelt H Optical-sideband cooling of visible atom

cloud confined in parabolic well Phys Rev Lett 41 233ndash236 (1978)7 Diedrich F Bergquist J C Itano W M amp Wineland D J Laser cooling to the zero-point energy of

motion Phys Rev Lett 62 403ndash406 (1989)8 Monroe C et al Resolved-side-band Raman cooling of a bound atom to the 3d zero-point energy

Phys Rev Lett 75 4011ndash4014 (1995)9 Kippenberg T J amp Vahala K J Cavity opto-mechanics Opt Express 15 17172ndash17205 (2007)10 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)11 Arcizet O et al Radiation-pressure cooling and optomechanical instability of a micromirror Nature

444 71ndash74 (2006)12 Gigan S et al Self-cooling of a micromirror by radiation pressure Nature 444 67ndash70 (2006)13 Schliesser A et al Radiation pressure cooling of a micromechanical oscillator using dynamical

backaction Phys Rev Lett 97 243905 (2006)14 Metzger C H amp Karrai K Cavity cooling of a microlever Nature 432 1002ndash1005 (2004)15 Naik A et al Cooling a nanomechanical resonator with quantum back-action Nature 443

193ndash196 (2006)16 Corbitt T et al An all-optical trap for a gram-scale mirror Phys Rev Lett 98 150802 (2007)

17 Thomson J D et al Strong dispersive coupling of a high finesse cavity to a micromechanicalmembrane Nature 452 72ndash75 (2008)

18 Brown K R et al Passive cooling of a micromechanical oscillator with a resonant electric circuitPhys Rev Lett 99 137205 (2007)

19 Wilson-Rae I Zoller P amp Imamoglu A Laser cooling of a nanomechanical resonator mode to itsquantum ground state Phys Rev Lett 92 075507 (2004)

20 Wineland D et al Experimental issues in coherent quantum-state manipulation of trapped atomicions J Res Natl Inst Standards Technol 103 259ndash328 (1998)

21 Tian L amp Zoller P Coupled ion-nanomechanical systems Phys Rev Lett 93 266403 (2004)22 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)23 Regal C A Teufel J D amp Lehnert K W Measuring nanomechanical motion with a microwave

cavity interferometer Preprint at lthttparXiv08011827gt (2008)24 Blencowe M P amp Buks E Quantum analysis of a linear dc SQUID mechanical displacement

detector Phys Rev B 76 014511 (2007)25 Martin I Shnirman A Tian L amp Zoller P Ground-state cooling of mechanical resonators

Phys Rev B 69 125339 (2004)26 Blencowe M P Imbers J amp Armour A D Dynamics of a nanomechanical resonator coupled to a

superconducting single-electron transistor New J Phys 7 236 (2005)27 Wilson-Rae I Nooshi N Zwerger W amp Kippenberg T J Theory of ground state cooling of a

mechanical oscillator using dynamical backaction Phys Rev Lett 99 093902 (2007)28 Marquardt F Chen J P Clerk A A amp Girvin S M Quantum theory of cavity-assisted sideband

cooling of mechanical motion Phys Rev Lett 99 093902 (2007)29 Armani D K Kippenberg T J Spillane S M amp Vahala K J Ultra-high-Q toroid microcavity on a

chip Nature 421 925ndash928 (2003)30 Carmon T et al Temporal behavior of radiation-pressure-induced vibrations of an optical

microcavity phonon mode Phys Rev Lett 94 223902 (2005)31 Kippenberg T J et al Analysis of radiation-pressure induced mechanical oscillation of an optical

microcavity Phys Rev Lett 95 033901 (2005)32 Rokhsari H Kippenberg T J Carmon T amp Vahala K J Radiation-pressure-driven

micro-mechanical oscillator Opt Express 13 5293ndash5301 (2005)33 Ma R et al Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres

Opt Lett 32 2200ndash2202 (2007)34 Hansch T W amp Couillaud B Laser frequency stabilization by polarization spectroscopy of a

reflecting reference cavity Opt Commun 35 441ndash444 (1980)35 Arcizet O et al High-sensitivity optical monitoring of a micromechanical resonator with a

quantum-limited optomechanical sensor Phys Rev Lett 97 133601 (2006)36 Poggio M Degen C L Mamin H J amp Rugar D Feedback cooling of a cantileverrsquos fundamental

mode below 5 mK Phys Rev Lett 99 017201 (2007)37 Raab C et al Motional sidebands and direct measurement of the cooling rate in the resonance

fluorescence of a single trapped ion Phys Rev Lett 85 538ndash541 (2000)38 Leibfried D Blatt R Monroe C amp Wineland D Quantum dynamics of single trapped ions Rev

Mod Phys 75 281ndash324 (2003)39 Dorsel A McCullen J D Meystre P Vignes E amp Walther H Optical bistability and mirror

confinement induced by radiation pressure Phys Rev Lett 51 1550ndash1553 (1983)40 Schwab K C amp Roukes M L Putting mechanics into quantum mechanics Phys Today 58

36ndash42 (2005)41 Mancini S Giovannetti V Vitali D amp Tombesi P Entangling macroscopic oscillators exploiting

radiation pressure Phys Rev Lett 88 120401 (2002)42 Kuzmich A Mandel L amp Bigelow N P Generation of spin squeezing via continuous quantum

nondemolition measurement Phys Rev Lett 85 1594ndash1597 (2000)

Supplementary Information accompanies this paper on wwwnaturecomnaturephysics

AcknowledgementsThe authors acknowledge discussions with T W Hansch W Zwerger and I Wilson-Rae TJKacknowledges support through an Independent Max Planck Junior Research Group Grant a MarieCurie Excellence Grant (JRG-UHQ) the DFG-funded Nanosystems Initiative Munich (NIM) and aMarie Curie Reintegration Grant (RG-UHQ) The authors gratefully acknowledge J Kotthaus foraccess to clean-room facilities for microfabrication and A Marx for support with scanningelectron microscopy

Author informationReprints and permission information is available online at httpnpgnaturecomreprintsandpermissionsCorrespondence and requests for materials should be addressed to TJK

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 419

copy 2008 Nature Publishing Group

ARTICLES

Frequency (MHz)

0 1

0ndash50

+50

Time (μs)

x (fm

)

times 025

Frequency (MHz)1495 1500 2855 2860

δx (1

0ndash18

m H

zndash12

)δx

(10ndash1

8 m

Hzndash1

2)

40604055

δx (1

0ndash18

m H

zndash1

2 )

31 2

3

12

3

73 74 73 74

Frequency (MHz)Frequency (MHz)

Γ eff2π = 16 MHz

Pcool = 0 mW

1

10

100

1000

0 10 20 30 40

100

200

300

0

10

100

δx (1

0ndash18

m H

zndash12

)

10

100Pcool = 27 mW

a

b

c

d

Figure 4 Resolved-sideband cooling of the radial breathing modea Time-domain trace of the brownian motion of the radial breathing mode asobserved by the monitoring laser in a 2 MHz spectral bandwidth aroundΩm2π= 406MHz For observation times that are short compared with thecoherence time of the mechanical oscillator a sinusoidal oscillation is observedb Full spectrum of the displacement fluctuation spectral density δx at roomtemperature recorded with the NdYAG laser (red) The different peaks appearing inthe spectrum represent the mechanical eigenmodes that can be identified usingthree-dimensional finite-element analysis The modes denoted by (123) arerotationally symmetric mechanical modes the strain (colour code) and deformedshapes of which are shown in the insets The background of the measurement (grey)is due to shot noise its frequency dependence results from the reduceddisplacement sensitivity (for the same measured noise level) at frequenciesexceeding the cavityrsquos bandwidth A signal-to-background ratio close to 60 dB in thenoise power spectral density is achieved at room temperature c Resolved-sidebandcooling with the cooling laser tuned to the lower sideband of the radially symmetricradial breathing mode (3) As evident only mode (3) is cooled whereas all othermodes (of which 1 and 2 are shown) remain unaffected Circles represent noisespectra with the cooling laser off (red) and running at 300 microW (blue) Lines arelorentzian fits d Cooling rates exceeding 15 MHz obtained with the 735 MHz radialbreathing mode of a different sample

emerging from the cavity (as analysed theoretically in ref 27) Toconfirm this aspect the motional sidebands generated during thecooling cycles were probed similar to spectroscopy of the resonancefluorescence of a cooled ion37 This is achieved with a heterodyne

Cool

ing

lase

r

BS

BS

AOM200 MHz

Heterodyne detection

Frequency (MHz)

15935 2406024055 240651594515940

PSD

(dB)

ndash10

ndash20

ω Lndash Ω AOMω L

ω L

ω L

plusmn ΩmΩ AOM

Ωm Ωm

a

b

-toroidμ

Figure 5 Motional sideband spectroscopy a Experimental set-up used to resolveupper and lower motional sidebands generated during interaction of the coolinglaser with the cavity similar to the spectroscopy of the resonance fluorescence of alaser-cooled ion37 The cooling laser interacts with the optical microcavity thetransmission of which is subsequently superimposed with a second laser beamretrieved from the same cooling laser but down-shifted by 200MHz using anacousto-optic modulator The beating of the two signals is recorded using a balancedheterodyne detector yielding spectral components of the lower and upper sidebandsat 200MHzminusΩm2π and 200MHz+Ωm2π respectively BS beam splitterb Beat signals of the upper (anti-Stokes) and lower (Stokes) motional sidebands for∆= 0 (red) and a detuning close to ∆sim= minusΩm (blue) The plotted electrical noisepower spectral density (PSD) is proportional to the optical power spectral density inthe sidebands For zero detuning of the pump with respect to the optical cavity themotional sidebands are equal in power By tuning the laser to the lower sidebandwhich induced cooling strong reduction of the Stokes sideband is observed

experiment by beating the cooling laser with a local oscillator(Fig 5a) derived by down-shifting part of the cooling laser lightusing an acousto-optic modulator at ΩAOM2π = 200 MHz Thebeat of the local oscillator and the cooling laser produces amodulation at ΩAOM whereas the motional sidebandsrsquo signals nowappear at ΩAOM plusmnΩm thereby enabling measuring their individualweights Figure 5b shows the result of this measurement for twodifferent laser detunings Whereas for excitation on the cavity linecentre (∆=0) the sideband intensities are equal (Aminusn asymp A+(n+1)because n 1 and Aminus

= A+) detuning the laser to the lowersideband ∆=minusΩm should lead to a strong suppression of the red-sideband beat by a factor of AminusA+ sim= 16Ω 2

mκ2 In the experimentwith the 406 MHz sample the detuning is chosen such thatthe red sideband is still discernible above the laser phase noisecorresponding to a suppression of more than 15 dB Optimizingthe laser detuning the red emission sideband could be reducedeven further It is important to note that the ability to measurethe individual sidebands separatelymdashas demonstrated heremdashisimportant for future experiments that venture into the quantumregime As theoretically predicted27mdashand in analogy to trappedions738mdashthe weights of the sidebands enable inferring the averagemotional occupation number27 for low occupancies by measuringthe ratio between the red and blue sidebands

The regime of resolved sidebands has another importantmdashand counterintuitivemdashbenefit because the cooling rate is indeedhigher in comparison with the unresolved case Keeping the

418 nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics

copy 2008 Nature Publishing Group

ARTICLES

launched power P as well as Ωm and R fixed an increase inthe cavity finesse increases the cooling rate until it saturates inthe highly resolved-sideband case and approaches an asymptoticvalue (see the Supplementary Information) The circulating powerhowever continues to decrease mitigating undesired effects suchas photothermal14- or radiation-pressure-induced bistability39 orabsorption-induced heating It is important to note however thatthe final occupancies for very high optical finesse are bound by thecavity decay rate28

Pertaining to the wider implications of our work we notethat resolved-sideband cooling as demonstrated here is a keyprerequisite for ground-state cooling and is an enabling stepfor both electromechanical and optomechanical schemes towardsobserving quantum mechanical phenomena of macroscopicobjects4041 such as the generation of non-classical states ofmotion Equally important the regime of resolved sidebandsenables a measurement scheme that permits measuring only onemotion quadrature To this end a lsquocontinuous two-transducermeasurementrsquo3 is required using a simultaneous excitation at boththe first upper and lower motional sidebands Theoretically thisscheme enables exceeding the standard quantum limit by

radicΩmκ

a factor of about 5 for the parameters reported here Alongthese lines continuous quantum non-demolition measurementsof a mechanical oscillator may produce squeezed states ofmechanical motion42

Received 20 January 2008 accepted 7 March 2008 published 13 April 2008

References1 Wineland D J amp Itano W M Laser cooling of atoms Phys Rev A 20 1521ndash1540 (1979)2 Stenholm S The semiclassical theory of laser cooling Rev Mod Phys 58 699ndash739 (1986)3 Braginsky V B amp Khalili F Y Quantum Measurement (Cambridge Univ Press Cambridge 1992)4 Dehmelt H G Entropy reduction by motional sideband excitation Nature 262 777 (1976)5 Wineland D amp Dehmelt H Proposed 10141υ lt υ laser fluorescence spectroscopy on

T1+mono-ion oscillator III Bull Am Phys Soc 20 637ndash637 (1975)6 Neuhauser W Hohenstatt M Toschek P amp Dehmelt H Optical-sideband cooling of visible atom

cloud confined in parabolic well Phys Rev Lett 41 233ndash236 (1978)7 Diedrich F Bergquist J C Itano W M amp Wineland D J Laser cooling to the zero-point energy of

motion Phys Rev Lett 62 403ndash406 (1989)8 Monroe C et al Resolved-side-band Raman cooling of a bound atom to the 3d zero-point energy

Phys Rev Lett 75 4011ndash4014 (1995)9 Kippenberg T J amp Vahala K J Cavity opto-mechanics Opt Express 15 17172ndash17205 (2007)10 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)11 Arcizet O et al Radiation-pressure cooling and optomechanical instability of a micromirror Nature

444 71ndash74 (2006)12 Gigan S et al Self-cooling of a micromirror by radiation pressure Nature 444 67ndash70 (2006)13 Schliesser A et al Radiation pressure cooling of a micromechanical oscillator using dynamical

backaction Phys Rev Lett 97 243905 (2006)14 Metzger C H amp Karrai K Cavity cooling of a microlever Nature 432 1002ndash1005 (2004)15 Naik A et al Cooling a nanomechanical resonator with quantum back-action Nature 443

193ndash196 (2006)16 Corbitt T et al An all-optical trap for a gram-scale mirror Phys Rev Lett 98 150802 (2007)

17 Thomson J D et al Strong dispersive coupling of a high finesse cavity to a micromechanicalmembrane Nature 452 72ndash75 (2008)

18 Brown K R et al Passive cooling of a micromechanical oscillator with a resonant electric circuitPhys Rev Lett 99 137205 (2007)

19 Wilson-Rae I Zoller P amp Imamoglu A Laser cooling of a nanomechanical resonator mode to itsquantum ground state Phys Rev Lett 92 075507 (2004)

20 Wineland D et al Experimental issues in coherent quantum-state manipulation of trapped atomicions J Res Natl Inst Standards Technol 103 259ndash328 (1998)

21 Tian L amp Zoller P Coupled ion-nanomechanical systems Phys Rev Lett 93 266403 (2004)22 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)23 Regal C A Teufel J D amp Lehnert K W Measuring nanomechanical motion with a microwave

cavity interferometer Preprint at lthttparXiv08011827gt (2008)24 Blencowe M P amp Buks E Quantum analysis of a linear dc SQUID mechanical displacement

detector Phys Rev B 76 014511 (2007)25 Martin I Shnirman A Tian L amp Zoller P Ground-state cooling of mechanical resonators

Phys Rev B 69 125339 (2004)26 Blencowe M P Imbers J amp Armour A D Dynamics of a nanomechanical resonator coupled to a

superconducting single-electron transistor New J Phys 7 236 (2005)27 Wilson-Rae I Nooshi N Zwerger W amp Kippenberg T J Theory of ground state cooling of a

mechanical oscillator using dynamical backaction Phys Rev Lett 99 093902 (2007)28 Marquardt F Chen J P Clerk A A amp Girvin S M Quantum theory of cavity-assisted sideband

cooling of mechanical motion Phys Rev Lett 99 093902 (2007)29 Armani D K Kippenberg T J Spillane S M amp Vahala K J Ultra-high-Q toroid microcavity on a

chip Nature 421 925ndash928 (2003)30 Carmon T et al Temporal behavior of radiation-pressure-induced vibrations of an optical

microcavity phonon mode Phys Rev Lett 94 223902 (2005)31 Kippenberg T J et al Analysis of radiation-pressure induced mechanical oscillation of an optical

microcavity Phys Rev Lett 95 033901 (2005)32 Rokhsari H Kippenberg T J Carmon T amp Vahala K J Radiation-pressure-driven

micro-mechanical oscillator Opt Express 13 5293ndash5301 (2005)33 Ma R et al Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres

Opt Lett 32 2200ndash2202 (2007)34 Hansch T W amp Couillaud B Laser frequency stabilization by polarization spectroscopy of a

reflecting reference cavity Opt Commun 35 441ndash444 (1980)35 Arcizet O et al High-sensitivity optical monitoring of a micromechanical resonator with a

quantum-limited optomechanical sensor Phys Rev Lett 97 133601 (2006)36 Poggio M Degen C L Mamin H J amp Rugar D Feedback cooling of a cantileverrsquos fundamental

mode below 5 mK Phys Rev Lett 99 017201 (2007)37 Raab C et al Motional sidebands and direct measurement of the cooling rate in the resonance

fluorescence of a single trapped ion Phys Rev Lett 85 538ndash541 (2000)38 Leibfried D Blatt R Monroe C amp Wineland D Quantum dynamics of single trapped ions Rev

Mod Phys 75 281ndash324 (2003)39 Dorsel A McCullen J D Meystre P Vignes E amp Walther H Optical bistability and mirror

confinement induced by radiation pressure Phys Rev Lett 51 1550ndash1553 (1983)40 Schwab K C amp Roukes M L Putting mechanics into quantum mechanics Phys Today 58

36ndash42 (2005)41 Mancini S Giovannetti V Vitali D amp Tombesi P Entangling macroscopic oscillators exploiting

radiation pressure Phys Rev Lett 88 120401 (2002)42 Kuzmich A Mandel L amp Bigelow N P Generation of spin squeezing via continuous quantum

nondemolition measurement Phys Rev Lett 85 1594ndash1597 (2000)

Supplementary Information accompanies this paper on wwwnaturecomnaturephysics

AcknowledgementsThe authors acknowledge discussions with T W Hansch W Zwerger and I Wilson-Rae TJKacknowledges support through an Independent Max Planck Junior Research Group Grant a MarieCurie Excellence Grant (JRG-UHQ) the DFG-funded Nanosystems Initiative Munich (NIM) and aMarie Curie Reintegration Grant (RG-UHQ) The authors gratefully acknowledge J Kotthaus foraccess to clean-room facilities for microfabrication and A Marx for support with scanningelectron microscopy

Author informationReprints and permission information is available online at httpnpgnaturecomreprintsandpermissionsCorrespondence and requests for materials should be addressed to TJK

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 419

copy 2008 Nature Publishing Group

ARTICLES

launched power P as well as Ωm and R fixed an increase inthe cavity finesse increases the cooling rate until it saturates inthe highly resolved-sideband case and approaches an asymptoticvalue (see the Supplementary Information) The circulating powerhowever continues to decrease mitigating undesired effects suchas photothermal14- or radiation-pressure-induced bistability39 orabsorption-induced heating It is important to note however thatthe final occupancies for very high optical finesse are bound by thecavity decay rate28

Pertaining to the wider implications of our work we notethat resolved-sideband cooling as demonstrated here is a keyprerequisite for ground-state cooling and is an enabling stepfor both electromechanical and optomechanical schemes towardsobserving quantum mechanical phenomena of macroscopicobjects4041 such as the generation of non-classical states ofmotion Equally important the regime of resolved sidebandsenables a measurement scheme that permits measuring only onemotion quadrature To this end a lsquocontinuous two-transducermeasurementrsquo3 is required using a simultaneous excitation at boththe first upper and lower motional sidebands Theoretically thisscheme enables exceeding the standard quantum limit by

radicΩmκ

a factor of about 5 for the parameters reported here Alongthese lines continuous quantum non-demolition measurementsof a mechanical oscillator may produce squeezed states ofmechanical motion42

Received 20 January 2008 accepted 7 March 2008 published 13 April 2008

References1 Wineland D J amp Itano W M Laser cooling of atoms Phys Rev A 20 1521ndash1540 (1979)2 Stenholm S The semiclassical theory of laser cooling Rev Mod Phys 58 699ndash739 (1986)3 Braginsky V B amp Khalili F Y Quantum Measurement (Cambridge Univ Press Cambridge 1992)4 Dehmelt H G Entropy reduction by motional sideband excitation Nature 262 777 (1976)5 Wineland D amp Dehmelt H Proposed 10141υ lt υ laser fluorescence spectroscopy on

T1+mono-ion oscillator III Bull Am Phys Soc 20 637ndash637 (1975)6 Neuhauser W Hohenstatt M Toschek P amp Dehmelt H Optical-sideband cooling of visible atom

cloud confined in parabolic well Phys Rev Lett 41 233ndash236 (1978)7 Diedrich F Bergquist J C Itano W M amp Wineland D J Laser cooling to the zero-point energy of

motion Phys Rev Lett 62 403ndash406 (1989)8 Monroe C et al Resolved-side-band Raman cooling of a bound atom to the 3d zero-point energy

Phys Rev Lett 75 4011ndash4014 (1995)9 Kippenberg T J amp Vahala K J Cavity opto-mechanics Opt Express 15 17172ndash17205 (2007)10 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)11 Arcizet O et al Radiation-pressure cooling and optomechanical instability of a micromirror Nature

444 71ndash74 (2006)12 Gigan S et al Self-cooling of a micromirror by radiation pressure Nature 444 67ndash70 (2006)13 Schliesser A et al Radiation pressure cooling of a micromechanical oscillator using dynamical

backaction Phys Rev Lett 97 243905 (2006)14 Metzger C H amp Karrai K Cavity cooling of a microlever Nature 432 1002ndash1005 (2004)15 Naik A et al Cooling a nanomechanical resonator with quantum back-action Nature 443

193ndash196 (2006)16 Corbitt T et al An all-optical trap for a gram-scale mirror Phys Rev Lett 98 150802 (2007)

17 Thomson J D et al Strong dispersive coupling of a high finesse cavity to a micromechanicalmembrane Nature 452 72ndash75 (2008)

18 Brown K R et al Passive cooling of a micromechanical oscillator with a resonant electric circuitPhys Rev Lett 99 137205 (2007)

19 Wilson-Rae I Zoller P amp Imamoglu A Laser cooling of a nanomechanical resonator mode to itsquantum ground state Phys Rev Lett 92 075507 (2004)

20 Wineland D et al Experimental issues in coherent quantum-state manipulation of trapped atomicions J Res Natl Inst Standards Technol 103 259ndash328 (1998)

21 Tian L amp Zoller P Coupled ion-nanomechanical systems Phys Rev Lett 93 266403 (2004)22 Braginsky V B Measurement of Weak Forces in Physics Experiments (Univ Chicago Press

Chicago 1977)23 Regal C A Teufel J D amp Lehnert K W Measuring nanomechanical motion with a microwave

cavity interferometer Preprint at lthttparXiv08011827gt (2008)24 Blencowe M P amp Buks E Quantum analysis of a linear dc SQUID mechanical displacement

detector Phys Rev B 76 014511 (2007)25 Martin I Shnirman A Tian L amp Zoller P Ground-state cooling of mechanical resonators

Phys Rev B 69 125339 (2004)26 Blencowe M P Imbers J amp Armour A D Dynamics of a nanomechanical resonator coupled to a

superconducting single-electron transistor New J Phys 7 236 (2005)27 Wilson-Rae I Nooshi N Zwerger W amp Kippenberg T J Theory of ground state cooling of a

mechanical oscillator using dynamical backaction Phys Rev Lett 99 093902 (2007)28 Marquardt F Chen J P Clerk A A amp Girvin S M Quantum theory of cavity-assisted sideband

cooling of mechanical motion Phys Rev Lett 99 093902 (2007)29 Armani D K Kippenberg T J Spillane S M amp Vahala K J Ultra-high-Q toroid microcavity on a

chip Nature 421 925ndash928 (2003)30 Carmon T et al Temporal behavior of radiation-pressure-induced vibrations of an optical

microcavity phonon mode Phys Rev Lett 94 223902 (2005)31 Kippenberg T J et al Analysis of radiation-pressure induced mechanical oscillation of an optical

microcavity Phys Rev Lett 95 033901 (2005)32 Rokhsari H Kippenberg T J Carmon T amp Vahala K J Radiation-pressure-driven

micro-mechanical oscillator Opt Express 13 5293ndash5301 (2005)33 Ma R et al Radiation-pressure-driven vibrational modes in ultrahigh-Q silica microspheres

Opt Lett 32 2200ndash2202 (2007)34 Hansch T W amp Couillaud B Laser frequency stabilization by polarization spectroscopy of a

reflecting reference cavity Opt Commun 35 441ndash444 (1980)35 Arcizet O et al High-sensitivity optical monitoring of a micromechanical resonator with a

quantum-limited optomechanical sensor Phys Rev Lett 97 133601 (2006)36 Poggio M Degen C L Mamin H J amp Rugar D Feedback cooling of a cantileverrsquos fundamental

mode below 5 mK Phys Rev Lett 99 017201 (2007)37 Raab C et al Motional sidebands and direct measurement of the cooling rate in the resonance

fluorescence of a single trapped ion Phys Rev Lett 85 538ndash541 (2000)38 Leibfried D Blatt R Monroe C amp Wineland D Quantum dynamics of single trapped ions Rev

Mod Phys 75 281ndash324 (2003)39 Dorsel A McCullen J D Meystre P Vignes E amp Walther H Optical bistability and mirror

confinement induced by radiation pressure Phys Rev Lett 51 1550ndash1553 (1983)40 Schwab K C amp Roukes M L Putting mechanics into quantum mechanics Phys Today 58

36ndash42 (2005)41 Mancini S Giovannetti V Vitali D amp Tombesi P Entangling macroscopic oscillators exploiting

radiation pressure Phys Rev Lett 88 120401 (2002)42 Kuzmich A Mandel L amp Bigelow N P Generation of spin squeezing via continuous quantum

nondemolition measurement Phys Rev Lett 85 1594ndash1597 (2000)

Supplementary Information accompanies this paper on wwwnaturecomnaturephysics

AcknowledgementsThe authors acknowledge discussions with T W Hansch W Zwerger and I Wilson-Rae TJKacknowledges support through an Independent Max Planck Junior Research Group Grant a MarieCurie Excellence Grant (JRG-UHQ) the DFG-funded Nanosystems Initiative Munich (NIM) and aMarie Curie Reintegration Grant (RG-UHQ) The authors gratefully acknowledge J Kotthaus foraccess to clean-room facilities for microfabrication and A Marx for support with scanningelectron microscopy

Author informationReprints and permission information is available online at httpnpgnaturecomreprintsandpermissionsCorrespondence and requests for materials should be addressed to TJK

nature physics VOL 4 MAY 2008 wwwnaturecomnaturephysics 419

copy 2008 Nature Publishing Group