Aquantumradiationpressurenoise-freeopticalspring W.ZachKorth,1 HaixingMiao,1,2 ThomasCorbitt,3 GarrettD.Cole,4 YanbeiChen,2 andRanaX.Adhikari1 1LIGO Laboratory, California Institute of Technology, Pasadena, CA 91125 2Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125 3Department of Physics & Astronomy, Louisiana State University, Baton Rouge, LA 70803 4ViennaCenterforQuantumScienceandTechnology(VCQ),FacultyofPhysics,UniversityofVienna,Austria (Dated:October2,2012) Recentadvancesinmicro-andnanofabricationhavespawnedgreatinterestinthefieldofcavityoptomechan- ics,whichprovidesapromisingavenuetowardsquantum-limitedmetrologyandstudiesofquantumbehaviors ofmacroscopicmechanicalobjects. Onemajorimpedimenttoreachingthequantumregimeisthermalexcita- tion,whichcanbeovercomeforsufficientlyhighmechanicalqualityfactorQ. Here,weproposeamethodfor increasingtheeffectiveQofamechanicalresonatorbystiffeningitviatheopticalspringeffectexhibitedby linearoptomechanicalsystems,andshowhowtheassociatedquantumradiationpressurenoisecanbeevaded 2 bysensingandfeedbackcontrol.Inaparameterregimethatiswithincurrenttechnology,thismethodallowsfor 1 realisticquantumcavityoptomechanicsinafrequencybandwellbelowthatwhichhasbeenrealizedthusfar. 0 2 Introduction.—Catalyzed by vast improvements in micro- chanical systems. If Q were truly a frequency-independent t c and nanofabrication processes, the field of cavity optome- quantity—as in the “structural damping” model proposed by O chanicshasseenaboomininterest[1–3]. Inadditiontopro- Saulson[15]—thenmovingtohighereigenfrequencieswould 1 viding a means for quantum-limited force measurements[4], leadtoanimmediateimprovement. Intheoppositedirection, e.g., in gravitational-wave detection[5] and scanning probe there are many experiments that would benefit from the use ] microscopy,optomechanicaldevicescanalsobeusedtoprobe of low-frequency (sub-kHz) resonators. A number of large h p thequantumbehaviorofmechanicalsystems. Recently, sev- bulk systems have been found to exhibit extremely high Q - eral experiments have demonstrated the cooling of a res- in this frequency range [16, 17]; unfortunately, these bulk t n onator down to its quantum ground state via cryogenics or systems tend to have relatively large (gram- to kg-scale) ef- a optomechanicalinteraction[6–8]. Inaddition,morethanone fective masses, making them unsuitable for typical optome- u group [9, 10] has demonstrated the so-called optomechani- chanicsexperiments. Therealizationofsub-microgrameffec- q cally induced transparency (OMIT) effect, an analog of the tivemassesrequirestheuseofmicrofabricatedresonators. In [ electromagnetically induced transparency (EIT) [11, 12] ef- practice, excess damping from surface effects [18], phonon 1 fectobservedinatomicsystems.Thiscanbeusedasanarrow- tunnelingloss[19]orintrinsicmechanismssuchasthermoe- v band quantum filter, e.g., to effect the frequency-dependent lastic[20]andAkhiezerdamping[21]limitstheachievableQ 9 0 phase rotation of squeezed light injected to enhance the sen- andthustheQf productforlow-frequencyresonators. Inad- 3 sitivityofquantumnoise-limitedinterferometricgravitational dition,weaddthefurtherrequirementthatthedesiredsystem 0 wave detectors[13, 14]. This also opens up the possibility exhibit excellent optical quality (i.e., high reflectivity owing . 0 for the processing and storing of nonclassical states of light tolowscatterlossandabsorption),whichlimitstheresonator 1 throughcoherenttransferofquantumstatesbetweenlightand options considerably, especially in light of the fact that typi- 2 amechanicaloscillator,atechniquethatwouldfindmuchuse caldielectricmaterialsusedtocreatemulti-layeropticalcoat- 1 intheemergentfieldofquantuminformationprocessing. ings (e.g., SiO /Ta O ) exhibit low mechanical quality fac- : 2 2 5 v Theubiquitousbathofthermalenergypresentsamajorob- tors [22]. Here, we propose a method for using the optical- i stacletotheseefforts,randomlyexcitingasystemandmask- spring effect in linear optomechanical devices[23–27] to in- X ing itsunderlying quantumnature. A characteristicfigure of creasetheeffectiveQofagivenmechanicalresonator,while r a meritforquantifyingthisthermaldecoherenceeffectisgiven simultaneouslyeliminatingtheexcessfluctuationduetoquan- bytheratioofthethermaloccupationnumbern¯ andtheme- tum radiation pressure noise imparted by the optical fields. th chanicalqualityfactorQ: This technique should facilitate the creation of an oscillator with a Qf product considerably higher than those available n¯ k T th = B ∝(Qf)−1, (1) today,enablingusefulapplicationsinquantummetrologyand Q h¯ωmQ alsocreationoflong-livedquantumstatesatlowerfrequencies thanwerepracticalbefore. where k is the Boltzmann’s constant and f =ω /2π is the B m mechanicalfrequency. Whenthisratiobecomessmallerthan Our idea uses the fact when a strong optical spring is one,theoscillatorquantumstatewillsurvivelongerthanone linearly coupled to a mechanical resonator, the resonator’s oscillationperiodbeforethethermaleffectdestroysit. hamiltonianbecomesaugmentedorevendominatedbycontri- As is apparent from Eq. (1), the quantum state lifetime butionsfromtheradiationpressureforcesoftheopticalfields. is ultimately limited by the product of the quality factor Q Inthisway,thebareresonator’sthermalnoiseis“diluted”by mechanical frequency f. A significant body of research has the ratio of the intrinsic elastic energy to that stored in the focused on increasing this product for a wide range of me- opticalfield[27]. Typically,themodificationofaresonator’s 2 dynamicsvialinearcouplingisaccompaniedbyexcessnoise from quantum back-action—the quantum fluctuation of the radiation pressure, in our case. This has been identified as a seriousissueinthestrongdilutionregimebyChangetal.[28] andNietal.[29],whoinsteadproposetoachieveopticaldi- lutionbyusinganonlinearquadraticopticalpotentialtotrap apartiallyreflectivemembrane[30],whichwouldbeimmune tolinearquantumback-action. Thedeviceweproposeevades such parasitic quantum back-action—i.e., the quantum radi- ationpressurenoise—byperforminghomodynedetectionon aproperoutgoingquadratureandactivelyfeedingbacktothe FIG.1: Simplifiedexperimentallayout,withthecanonicaloptome- oscillator,resultinginanearlynoise-freeopticalspring.Since chanicalsystemshownwithinthedashedbox. Inputvacuumfluctu- thismethodallowsforeasycouplingofthedilutedmechani- ationsdrivethecavitymode,whichinturnexertsradiationpressure calresonatortoanexternalopticalsystemfromtheotherside forcesonthemechanicalresonatorformingthecavityboundary.The of the resonator, it can be used as a black-box effective me- outputmodeofthecavityissensedviahomodynedetection,and,by measuringtheappropriatequadraturethatismentionedinthetext,an chanicalresonatorofexceptionallyhighmechanicalquality. errorsignalmaybederivedforfeedingbacktotheresonatorposition Optical spring.—The canonical optomechanical system is tocanceltheradiationpressurenoise. shown in the dashed box in Fig. 1. In such a system, the “optical spring” effect arises from dynamical back-action of the optical cavity field on the mechanical oscillator forming onecavityboundary. Themechanicaloscillatordisplacement fromitsoriginalvalue xˆ is coupled to the cavity field aˆ via radiation pressure, as R−1(ω)=−m(ω2+iγ ω−ω2) (5) describedbythefollowinginteractionHamiltonian[31]: 0 m m toaneffectiveone: Hˆ =h¯G¯ xˆ(aˆ†+aˆ)≡xˆFˆ . (2) int 0 rad R−1(ω)=−m[ω2+i(γ +Γ )ω−(ω2+ω2)]. (6) eff m os m os ThecouplingconstantisG¯ =a¯ω /L;a¯istheclassicalmean 0 c For a strong optical spring ω (cid:29)ω , we can significantly amplitude of aˆ due to coherent driving of an external laser; os m stiffen the mechanical oscillator with the restoring energy ω is the cavity resonant frequency; L is the cavity length. 0 fromtheopticalfield. When the frequency of the external laser ω that drives the l cavityfieldisdetunedfromω ,Fˆ dependsontheoscillator One immediate issue with this approach comes from the c rad quantum radiation pressure noise Fˆ (ω) in Eq.(3) which displacement, creating a mechanical response that mimics a noise spring. Morespecifically,Fˆ inthefrequencydomaincanbe arisesfromquantumfluctuationoftheopticalfield: rad √ writtenas(thedetailsofthederivationareintheAppendix): 2h¯G¯ γ[(γ−iω)vˆ (ω)+∆vˆ (ω)] Fˆ (ω)= 0 1 2 , (7) noise (ω+∆+iγ)(ω−∆+iγ) Fˆ (ω)=−K (ω)xˆ(ω)+Fˆ (ω), (3) rad os noise √ √ where vˆ ≡(aˆ +aˆ†)/ 2 and vˆ ≡(aˆ −aˆ†)/ 2i are the 1 in in 2 in in wheretheopticalspringcoefficientK isapproximatelygiven os amplitude and phase quadratures of the input optical field. by Thisadditionalnoisetermwillincreasetheeffectivetemper- atureofthethermalbath,anddrivethemechanicaloscillator 2h¯G¯2∆ 4ih¯G¯2γ∆ω K (ω)≈− 0 − 0 ≡mω2 −imΓ ω (4) away from the quantum regime, as pointed out by Chang et os ∆2+γ2 (∆2+γ2)2 os os al.[28]. Itsstrengthcanbequantifiedbythespectraldensity with the cavity detuning ∆≡ω0−ωl and γ being the cavity 4h¯2G¯2γ(γ2+ω2+∆2) 4h¯2G¯2γ bandwidth. Here, the approximation is taken for the case of SF(ω)= [(ω−∆)20+γ2][(ω+∆)2+γ2] ≈ γ2+∆02, (8) large detuning and cavity bandwidth, which we will show to be the relevant parameter regime for realization of this idea. wherewehaveagainassumedlargebandwidthanddetuning. In addition, we have introduced the optical spring frequency AswecanseefromtheaboveexpressionandEq.(4),theop- ωos and the optical damping Γos. As we can see, when the ticalrigidity(realpartofKos)scaleswiththeoptomechanical detuning is negative, i.e., ∆<0, the optical rigidity is real coupling strength in the same way as the quantum radiation and positive, and the optical damping is negative Γ (heat- pressurenoise: os ing), and vice versa. By introducing an additional driving K ,S ∝G¯2. (9) field with a different detuning frequency, we can create the os F 0 so-called stable double optical spring[26], which can make Essentially, this means that an increase in the optical spring boththerigidityandthedampingpositive(wewillelaborate frequencyisaccompaniedbyanincreaseintheradiationpres- on this issue later). The optical spring modifies the mechan- surenoisewhenwescaleuptheopticalpowerofthedriving ical response R (ω), defined through R (ω)≡xˆ(ω)/Fˆ(ω), laser. 0 0 3 Evading quantum radiation pressure noise.—To solve the can evade the quantum radiation pressure noise, as shown aforementioned issue, we make use of the fact that the out- schematicallyinFig.1. putfieldemergingfromthecavitycontainsinformationabout Residualradiationpressurenoise.—Whilestrongradiation the quantum radiation pressure noise that has been imposed pressure noise cancellation can be achieved using this tech- onto the mechanical oscillator. By choosing an appropriate nique, a small fraction cannot be canceled owing to two ef- quadrature of the output field, we can measure most of the fects:(i)opticallosswithinthecavityorfromnon-unityquan- radiation pressure noise without being sensitive to the oscil- tum efficiency in photodetection, which will introduce vac- lator displacement. By feeding this measurement outcome uum noise that is uncorrelated with vˆ and vˆ ; (ii) frequency 1 2 back to the mechanical oscillator with the proper filter, we dependence of the radiation pressure noise, which we have canevadethequantumradiationpressurenoiseandachievea thusfarignoredbyassumingalargedetuningandcavityband- nearly noise-free optical spring. Note that this does not vio- width. Inrealexperiments,thereisalwayscertainamountof latethefundamentalprincipleofquantummeasurement—any opticalloss,andthebandwidthanddetuningareallfinite. We linearcontinuousmeasurementofanydynamicalvariablethat canmakeanestimateofthemagnitudeoftheresidualnoise; doesnotcommuteatdifferenttimes(non-conservative)isas- supposing we cancel the leading-order term [c.f. Eq.(15)] sociatedwithquantumback-actiononthatvariable[4]. Here from the full radiation pressure force [c.f. Eq.(7)], there re- weonlysensethequantumradiationpressurenoiseandhave maintermsofO(ω)andhigher: essentiallynosensitivitytothemechanicaldisplacement,and √ thatiswhywecanevadesuchback-actionnoise. 2ih¯G¯ γ[(γ2−∆2)vˆ (ω)+2γ∆vˆ (ω)] Fˆres (ω)= 0 1 2 ω+O(ω2). Weelaborateonthisideausingtheinput-outputrelationfor noise (γ2+∆2)2 thissystem (16) (cid:112) Accountingforthis,andfortheimperfectcancellationofthe aˆ (ω)=−aˆ (ω)+ 2γaˆ(ω). (10) out in first term due to the finite loss ε, the power spectrum of the In terms of amplitude and phase quadratures: yˆ ≡(aˆ + residualradiationpressurenoiseis √ √ 1 out aˆ† )/ 2andyˆ ≡(aˆ −aˆ† )/ 2i,wehave(againforlarge out 2 out out 4h¯2G¯2γ (cid:18) ω2 (cid:19) detuningandbandwidth): Sres(ω)= 0 ε+ +O(ω4) . (17) √ F γ2+∆2 γ2+∆2 yˆ (ω)≈ (γ2−∆2)vˆ1(ω)+2γ∆vˆ2(ω)−2 γG¯0∆xˆ(ω), 1 ∆2+γ2 ∆2+γ2 Here, ε is the sum of the intracavity roundtrip loss and the (11) overalllossinphotodetectionduetoimperfectmodematching yˆ (ω)≈ (γ2−∆2)vˆ2(ω)−2γ∆vˆ1(ω)−2√γG¯0γxˆ(ω). aqnudalnitoyno-ufncituyrrqeunatlnytuamvaeiflfiabclieenocpytiocfatlheelpehmoetnotdsi,oidtew(gililvelinktehlye 2 ∆2+γ2 ∆2+γ2 belimitedbyquantumefficiencyatalevelofε ≈0.1−1%). (12) Sincethefrequencyweareinterestedinisaroundtheshifted By choosing the appropriate local oscillator phase, we can mechanical resonant frequency, ωos, the effect of the finite perform homodyne detection on the following quadrature, bandwidthcanbenegligiblysmall. which is a linear combination of the above output amplitude Bycomparisonwiththethermalforcespectrumfromavis- and phase quadratures (yˆζ ≡yˆ1sinζ +yˆ2cosζ with tanζ = cousdampingmodel,SFth=4mγmkBT,wecanassignaneffec- −γ/∆): tivetemperaturetosuchresidualnoise,as yˆζ(ω)= γvˆ1((∆ω2)++γ∆2)vˆ12/(2ω) ≡vˆζ(cid:48)(ω). (13) Terfefs≡ 4mSγFresk . (18) m B This quadrature has nearly no response to the oscillator dis- This effective temperature must be below the required bath placementxˆ, whileitcontainsaparticularquadraturevˆζ(cid:48)(ω) temperatureforagivenapplication. ofthequantumfluctuation: Experimental realization with double optical spring.—As vˆζ(cid:48)(ω)≡vˆ1(ω)sinζ(cid:48)+vˆ2(ω)cosζ(cid:48), (14) mentioned earlier, a single optical spring is unstable by na- ture[cf. Eq.(4)]—inthecaseofablue-detunedfield,thesys- with tanζ(cid:48) =γ/∆. This turns out to be the same quadrature temexhibitsapositiverestoringforcebutanegativedamping responsible for the quantum radiation pressure noise, which force, whereas, for a red-detuned field, the forces are damp- canbeseenfromtheexpressionforFˆnoise [cf. Eq.(7)]andin ing but anti-restoring. Thus, a system whose dynamics are thelargedetuningandbandwidthcase: dominatedbyanopticalspringwillbeinherentlyunstable. 2h¯G¯ √γ[γvˆ (ω)+∆vˆ (ω)] AnovelapproachproposedinRef.[26]usesasecondopti- Fˆnoise(ω)≈ 0 (∆21+γ2) 2 ∝vˆζ(cid:48)(ω). (15) calspringfieldtocreateapassivelystablesystem. Thelinear combinationoftwoK s,withonered-detunedandtheother os Therefore, by feeding the measurement result of yˆ back to blue,canbemadetoexhibitbothpositiverestoringanddamp- ζ the mechanical oscillator with the correct linear filter, we ing,resultinginapassivelystablespring. Thesumofthetwo 4 opticalspringcontributionsisthus: should not be attempted, but it serves to illustrate what this technique implies in the context of quantum experiments)! (cid:34) (cid:35) ∆ ω2 ∆ ω2 From Eq.(18), we can also calculate the effective tempera- Kotost≈imω (γ2B+o∆sB2)+(γ2R+o∆sR2) +mωo2sB−mωo2sR tures of the residual quantum radiation pressure noise from B B R R thetwoopticalspringfieldsasTres,B≈0.5KandTres,R≈12 (19) eff eff K,inthelosslesscase,orTres,B≈10KandTres,R≈20K,for whereγB,γRand∆B,∆Rarethecavitybandwidthanddetuning eff eff ε=0.001. Eveninthelossycase,theresidualnoisetempera- asseenbytheblueandredfields,respectively(notethat∆ < B turesareconsiderablylowerthanareasonabletargetenviron- 0). For a proper choice of these parameters as a function of menttemperature. theratioω /ω >1,theexpressioninthebracketscanbe madetovaonsiBsh,aonsAdtheeffectiveresonatorisstiffenedwithout Experimental setup.—A possible experimental layout is instabilityorexcessdamping1 showninFig.2.Alaser’sfrequencyisstabilizedtotheoptical springcavitylengthusingthePound-Drever-Hall(PDH)lock- A set of sample parameters is given in Table I. Under ingtechnique. Anoffsetisinjectedintotheloopsuchthatthe these conditions, an oscillator with a resonant frequency of ω /2π≈100kHzandaneffectiveQof109isformed2.Such opticalfrequencyisred-detunedfromthecavityresonanceby os |∆ |.Apickoffbeamispassedthroughanacousto-opticmod- a device can in principle be cooled to its ground state from R ulator(AOM),whichupshiftsitsfrequencyby∆ +∆ ,such an environmental temperature of T ≈4800 K (Clearly, this R B thatitisblue-detunedfromthecavityresonanceby|∆ |. The B twobeamsareresonantinorthogonalpolarizationsforsimple isolation. Further pickoffs are taken from each path to be used as optical local oscillators for homodyne detection of their re- spective reflected beams. The photodetector signals are dif- ferenced at the appropriate homodyne angle, and these sig- nals are summed and filtered for feedback to the laser and AOM frequencies, which is done in such a way as to cancel themeasuredradiationpressurenoise.Inpractice,thetransfer functionfromlaserfrequencytoforceontheresonatorcanbe quitecomplicated. Inthiscase,amoredirectamplitudemod- ulationtechniquemightbeemployed(e.g.,byplacingAOMs inAMconfigurationonthebeamsenteringthecavity). Conclusion.—We have proposed a method for creating an effective mechanical resonator with far higher Qf product thananyavailabletoday. Inaddition,theseresonatorscanbe FIG.2: Theproposedexperimentalsetup. AlaserisPound-Drever- madetooperateinlowerfrequencybandsthancurrentonesof Hall (PDH) locked to the optical spring cavity via feedback to the competitivequality. Whiletheuseofopticaldilutiontomit- laserfrequency,andthenanoffset−|∆R|/2π isinjectedtodetuneit igate thermal noise has been proposed and demonstrated in totheredsideofresonance.Apickoffofthisbeamisupshiftedwith thepast,wehaveconsideredaparameterregimeinwhichthe anacousto-opticmodulator(AOM)by fAOM=(|∆R|+|∆B|)/2π,and deleterious effects of quantum radiation pressure noise from thenfedintothecavitywithanorthogonalpolarization,formingthe the strong optical spring fields can be all but eliminated, al- blue-detuned field. Homodyne detection is performed on each re- flected field at the appropriate angle, and a filtered combination of lowinginprinciplefordilutiontoarbitrarilyhighquality. We thesetwosignalsisappliedtoeachthelaserfrequencyandtheAOM feelthattheapplicationofthistechniqueholdsgreatpromise frequency,providingactivecancellationofradiationpressurenoise. foranyfieldrequiringvery-high-Qresonators,including,but notlimitedto,thoseofquantumoptomechanicsandsensitive forcemeasurement. The authors would like to gratefully acknowledge support from the National Science Foundation. Specifically: W.Z.K. 1Notethattheexpressionneednotvanish,butonlybepositiveforthere- andR.X.A.aresupportedbyNSFGrantPHY-0757058;H.M. sultantresonatortobestable.Furthermore,anypositivedampingfromthe andY.C.aresupportedbyNSFGrantsPHY-1068881andCA- opticalfieldsiscold, andthereforedoesnotcontributenoiseordegrade REER Grant PHY-0956189; T.C. is supported by NSF CA- SNR.Wespecificallyconsiderthecaseofzeroadditionaldamping,how- ever,sinceitleadstoaneffectiveresonatorwhoseQisdeterminedsolely REERGrantPHY-1150531. bytheintrinsicdampingofthebaremechanicalsystem. 2ThisQeffvalueiscalculatedassumingaviscousdampingmodel;theme- chanicaldamping, γm, isfixed, andso, sincetheopticalspringaddsno damping,theimprovementisgivenbyQeff=(ωos/ωm)Q. 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MacDonald, Sensors and Actua- iop.org/1367-2630/12/i=8/a=083032. tors A: Physical 50, 199 (1995), ISSN 0924-4247, URL [35] G.J.MilburnandM.J.Woolley,ActaPhysicaSlovaca61,483 http://www.sciencedirect.com/science/article/ (2011). pii/0924424795010807. 6 Appendix Similarly, we canread off the equation of motionfor the os- cillator: Inthisappendix, wewillshowsomeadditionaldetailsfor m[x¨ˆ(t)+γ x˙ˆ(t)+ω2xˆ(t)]=Fˆ (t)+Fˆ (t). (24) the derivation of the formulas presented in the main text, m m rad th which have been mostly covered in the literature[31–35]. Herewehavedefinedtheradiationpressure Here,weuseanotationnearlyidenticaltothoseinRef.[34]. WestartwiththestandardHamiltonianforthecanonicalop- Fˆ (t)≡−h¯G¯ [aˆ(t)+aˆ†(t)]. (25) rad 0 tomechanicaldevice,showninthedashedboxinFig.1: Inadditionwehaveaddedthedampingtermmγ x˙ˆ(t)andthe m Hˆ = pˆ2 +1mω2xˆ2+h¯ω aˆ†aˆ+h¯G xˆaˆ†aˆ associated thermal fluctuation force Fˆth into the equation of 2m 2 m c 0 motion, of which the correlation function is (cid:104)Fˆ (t)Fˆ (t(cid:48))(cid:105)= th th +ih¯(cid:112)2γ[aˆin(t)e−iω0taˆ†−aˆ†in(t)eiω0taˆ]. (20) 4mγmkBTδ(t−t(cid:48))inthehigh-temperaturelimitkBT (cid:29)h¯ωm. Solutiontothecavitymode.—Theabovelinearequationsof Here, thefirsttwotermsarethefreeHamiltonianfortheos- motioncanbesolvedinthefrequencydomain. Thesolution cillator with ωm being the mechanical frequency; the third forthecavitymodereads: term is the free Hamiltonian for the cavity mode, ωc is the √ cavity resonant frequency, and aˆ is its annihilation operator aˆ(ω)= G¯0xˆ(ω)+i 2γaˆin(ω). (26) satisfying [aˆ,aˆ†]=1; the fourth term describes the interac- ω−∆+iγ tion between the oscillator and the cavity mode, with G = 0 Fromthis,wecanobtaintheexpressionfortheradiationpres- ω /L being the coupling strength and L the cavity length; c sure: the remaining part is the coupling between the cavity mode with the external continuum aˆin(t), with coupling rate γ and Fˆ (ω)=−K (ω)xˆ(ω)+Fˆ (ω). (27) [aˆ (t),aˆ†(t(cid:48))]=δ(t−t(cid:48)). We have ignored those terms ac- rad os noise in in countingforthedissipationmechanismofthemechanicalos- WeintroducetheopticalspringcoefficientK as: os cillator coupling to its thermal environment. Later we will includetheireffectsintheequationofmotionfortheoscilla- K (ω)≡ 2h¯G¯20∆ , (28) os tor. (ω−∆+iγ)(ω+∆+iγ) Linearized Hamiltonian.—In the experiment, the cavity andthequantumradiationpressurenoisetermas: modeisdrivencoherentlybyalaserwithalargeamplitudeat frequencyω0.Wecanthereforestudythelinearizeddynamics 2h¯G¯ √γ[(γ−iω)vˆ (ω)+∆vˆ (ω)] perturbingaroundthesteadystate.Intherotatingframeofthe Fˆnoise(ω)≡ 0(ω−∆+iγ)(ω1 +∆+iγ)2 (29) laserfrequencyω ,thecorrespondinglinearizedHamiltonian 0 √ √ forthesystemreads: withvˆ ≡(aˆ +aˆ†)/ 2andvˆ ≡(aˆ −aˆ†)/ 2ibeingthe 1 in in 2 in in pˆ2 1 vacuumfluctuationoftheinputamplitudeandphasequadra- Hˆ =2m+2mωm2xˆ2+h¯∆aˆ†aˆ+h¯G¯0xˆ(aˆ†+aˆ) tures. The strength of the radiation-pressure noise can be +ih¯(cid:112)2γ[aˆ (t)aˆ†−aˆ†(t)aˆ], (21) quantifiedbyitspowerspectrumwhichisdefinedthrough in in (cid:104)0|Fˆ† (ω)Fˆ (ω(cid:48))|0(cid:105) ≡πS (ω)δ(ω−ω(cid:48)), (30) wherethecavitydetuningisthedifferencebetweenthecavity noise noise sym F resonantfrequencyandthelaserfrequency,i.e.,∆≡ωc−ω0, wherethesubscript‘sym’denotesforsymmetrizationandthe the linear coupling strength is G¯0 = G0a¯ with a¯ being the spectrumisasingle-sidedone. Noticethatforvacuuminput steady-state amplitude of the cavity mode. These operators state(cid:104)0|vˆ†(ω)vˆ(ω(cid:48))|0(cid:105) =πδ δ(ω−ω(cid:48)),andtherefore k l sym kl should be viewed as perturbed parts of the original ones and thequantumstatetheyactonisalsotransformedcorrespond- 4h¯2G¯2γ(γ2+ω2+∆2) S (ω)= 0 . (31) ingly. For instance, the input state for aˆin is originally a co- F [(ω−∆)2+γ2][(ω+∆)2+γ2] herentstate(foranideallaser),andnowitisthevacuumstate |0(cid:105)with(cid:104)0|aˆ (t)aˆ†(t(cid:48))|0(cid:105)=δ(t−t(cid:48)). Forthecaseoflargebandwidthanddetuningthatwearein- in in Equations of motion.—Given the above Hamiltonian, the terestedin,theaboveradiationpressurenoisecanbeapproxi- cavity mode satisfies the following Heisenberg equation of matedas(uptozerothorderofω): motion: √ 2h¯G¯ γ[γvˆ (ω)+∆vˆ (ω)] a˙ˆ(t)+(γ+i∆)aˆ(t)=−iG¯ xˆ(t)+(cid:112)2γaˆ (t). (22) Fˆnoise(ω)≈ 0 ∆21+γ2 2 . (32) 0 in anditisrelatedtothecavityoutputaˆ bythestandardinput- This indicates that only a particular linear combination — out outputrelation: γvˆ1+∆vˆ2 — of the amplitude and phase quadrature fluctu- ationisresponsibleforthequantumradiation-pressurenoise. (cid:112) aˆ (t)=−aˆ (t)+ 2γaˆ(t). (23) It turns out that we such a combination is measurable in the out in 7 cavityoutput,andthatiswhywecanevadethequantumradi- Itisconvenienttoreexpressitintermsofamplitudeandphase ation pressure noise by feeding back the measurement result quadratures, ofwhichthelinearcombinationismeasuredby withanappropriatelinearfilter,whichisthemainideaofthis usingahomodynedetectionscheme.Fortheoutputamplitude work. quadrature,wehave Solution to the mechanical oscillator.—Given the expres- sionfortheradiationpressure,wecanwritedownthesolution aˆ (ω)+aˆ† (−ω) yˆ (ω)≡ out √ out forthemechanicaldisplacementxˆas: 1 2 xˆ(ω)= Fˆnoise(ω)+Fˆth(ω) . (33) =−(ω2(−ω∆−2+∆+γ2i)γvˆ)1((ωω)++∆2+γ∆iγvˆ)2(ω) −m[ω2−ω2+iγ ω]+K (ω) m m os √ 2 γG¯ ∆xˆ(ω) 0 + . (38) As we can see, the mechanical response is modified into an (ω−∆+iγ)(ω+∆+iγ) effective one due to the optical-spring effect. Since we are Forthephasequadrature,wehave focusinginlargecavitybandwidthanddetuningcase,theop- ticalspringK canbeexpandedas: os aˆ (ω)−aˆ† (−ω) yˆ (ω)≡ out √ out 2h¯G¯2∆ 4ih¯G¯2γ∆ω 2 2i K ≈− 0 − 0 ≡mω2 −imΓ ω, (34) os ∆2+γ2 (∆2+γ2)2 os os =−(ω2−∆2+γ2)vˆ2(ω)−2γ∆vˆ1(ω) (ω−∆+iγ)(ω+∆+iγ) whereωos istheoptical-springfrequencyandΓos istheopti- 2√γG¯ (γ−iω)xˆ(ω) caldampingcoefficient. Wecanthenrewritethemechanical + 0 . (39) displacementxˆas (ω−∆+iγ)(ω+∆+iγ) xˆ(ω)=R−1(ω)[Fˆ (ω)+Fˆ (ω)], (35) Giventhelargebandwidthanddetuning,wecanapproximate eff noise th themas: fiwnheedrethtrhoeuegfhf:ective mechanical response function Reff is de- yˆ (ω)≈ (γ2−∆2)vˆ1(ω)+2γ∆vˆ2(ω)−2√γG¯0∆xˆ(ω), 1 ∆2+γ2 ∆2+γ2 R−1(ω)≡−m[ω2+i(γ +Γ )ω−(ω2+ω2)]. (36) (40) eff m os m os √ Inthenegativedetuningcase∆<0, ωos ispositiveandreal, yˆ2(ω)≈ (γ2−∆2)vˆ∆2(2ω+)γ−22γ∆vˆ1(ω)−2∆2γ+G¯γ02γxˆ(ω). andthedampingΓ isnegative; whileinthepositivedetun- os (41) ing case ∆>0, ω is pure imaginary and the damping Γ os os is positive. In both cases, the mechanical system is poten- The linear combination of the above quadratures that con- tially unstable, especially when the intrinsic damping γ is m tains the fluctuation responsible for quantum radiation pres- small as in our proposed parameter regime. By introducing surenoise[cf. Eq.(29)]is: anadditionallaserthathasadifferentdetuningfrequency,we cancombinetwoopticalspringandachievebothpositivefre- yˆ (ω)≡sinζyˆ (ω)+cosζyˆ (ω), (42) quency and damping — the so-called double optical spring. ζ 1 2 SuchaschemehasbeenrealizedexperimentallybyCorbittet withtanζ =−γ/∆,whichgives: al.[26]. Wecanthereforesignificantlyupshiftthemechanical resonantfrequencywhilekeepingtheoscillatorstable. γvˆ (ω)+∆vˆ (ω) Solutiontothecavityoutput.—Fromtheinput-outputrela- yˆ (ω)= 1 2 . (43) ζ (∆2+γ2)1/2 tion,thecavityoutputisgivenby: ω−∆−iγ √2γG¯ By measuring this quadrature, we will be able to evade the aˆ (ω)=− aˆ (ω)+ 0 xˆ(ω). (37) quantumradiationpressurenoise. out ω−∆+iγ in ω−∆+iγ