Synchronization of an optomechanical system to an external drive

Transcription

1 Sycroizatio of a optomcaical systm to a xtral driv Eud Amitai, 1 Nils Lörc, 1 Adras Nukamp, Stfa Waltr, 3 ad Cristop Brudr 1 1 Dpartmt of Pysics, Uivrsity of Basl, Kliglbrgstrass 8, 4056 Basl, Switzrlad Cavdis Laboratory, Uivrsity of Cambridg, Cambridg CB3 0HE, Uitd Kigdom 3 Max Plack Istitut for t Scic of Ligt, Staudtstrass, Erlag, Grmay Optomcaical systms driv by a ffctiv blu dtud lasr ca xibit slf-sustaid oscillatios of t mcaical oscillator. Ts slf-oscillatios ar a prrquisit for t obsrvatio of sycroizatio. Hr, w study t sycroizatio of t mcaical oscillatios to a xtral rfrc driv. W study two cass of rfrc drivs: (1) A additioal lasr applid to t optical cavity; () A mcaical driv applid dirctly to t mcaical oscillator. Startig from a mastr quatio dscriptio, w driv a microscopic Adlr quatio for bot cass, valid i t classical rgim i wic t quatum sot ois of t mcaical slfoscillator dos ot play a rol. Furtrmor, w umrically sow tat, i bot cass, sycroizatio ariss also i t quatum rgim. T optomcaical systm is trfor a good cadidat for t study of quatum sycroizatio. PACS umbrs: Xt, k, Cm I. INTRODUCTION Sycroizatio is t pomo i wic a limit-cycl oscillator, i.., a oscillator wit fixd amplitud ad fr pas, dvlops a pas prfrc w wakly coupld to a driv or to otr slf-oscillatig systms 1, ]. Tis pomo is prvalt i all t atural scics, maifstig itslf i, for xampl, adjustmt of t circadia rytm i may livig systms or firflis blikig i uiso 3]. I rct yars tr as b cosidrabl itrst i t topic of quatum sycroizatio 4 1], i.., t sycroizatio of slf-oscillators opratig i t quatum rgim. Tr as b xtsiv rsarc do o t paradigmatic xampl of a va dr Pol oscillator 4 1]. Otr platforms av b usd to study quatum sycroizatio as wll, amog wic ar micromasrs 13], atomic smbls 14, 15], itractig quatum dipols 16], trappd ios 6, 17], ad optomcaical systms 18 0]. I a optomcaical systm lctromagtic cavity mods ar coupld to mcaical motio. I its most basic stup, a optomcaical systm is mad of a sigl lasr-driv cavity mod wic coupls to a sigl mcaical mod via,.g., radiatio prssur ]. T dyamics of t systm crucially dpds o t frqucy of t lasr drivig t cavity. A lasr fild tud to t rd sid of t cavity frqucy is usd for back-actio coolig as wll as for stat trasfr 3 5], wil rsoat drivig is usd,.g., for positio ssig 6]. W blu dtud, t lasr driv ca iduc t paramtric istability, ladig i tur to slf-sustaid oscillatios of t mcaical oscillator. Ts slf-sustaid oscillatios av b studid i bot t classical ad t quatum rgims 7 3]. For tat raso, t optomcaical systm may xibit sycroizatio w coupld to a xtral driv (a additioal xtral driv, i cotrast to t lasr drivig t slf-oscillatios), to aotr optomcaical systm, or as part of a array of optomcaical systms, as was tortically sow i t classical rgim 33, 34]. Sycroizatio of a optomcaical systm to a xtral driv 35], of two optomcaical systms 36] ad v of small arrays of up to sv suc systms 37] av b dmostratd xprimtally. I t quatum rgim t sycroizatio of two optomcaical systms as b studid tortically 18], as wll as t sycroizatio of a array of suc systms 19] witi a ma-fild approac was usd. I tis work, w tortically study t sycroizatio of t mcaical slf-oscillator to a xtral rfrc driv. W xami two diffrt rfrc drivs: (1) A additioal lasr applid to t optical cavity. Udr a appropriat rotatig-wav approximatio, tis may also b implmtd by modulatig t powr of t lasr iducig t mcaical slf-oscillatios, as was xprimtally do i Rf. 35]; () A mcaical driv applid dirctly oto t mcaical oscillator, wic could for istac b ralizd wit a pizolctric lmt attacd to t mcaical oscillator. For bot cass, our startig poit of t aalysis is t microscopic mastr quatio. W t us t lasr tory for optomcaical limit cycls 30] to driv a quatio of motio (EOM) for t pas distributio of t mcaical slfoscillator. W sow tat i bot cass, i a rlvat paramtr rgim, t Adlr quatio mrgs from t EOM. T Adlr quatio is a diffrtial quatio for t pas diffrc btw t slf-oscillator ad t rfrc driv, kow to dscrib sycroizatio. For t optical rfrc driv, tis is t first tim a microscopically drivd Adlr quatio is discussd. For t mcaical rfrc driv, it rproducs a rsult i Rf. 33]. W t cotiu to sow umrically, for bot cass, tat i t quatum paramtr rgim a Arold togu xists, a stadard sigatur of sycroizatio 1, ]. Tis suggsts t optomcaical systm is a good cadidat for t study of sycroizatio i t quatum rgim. Tis mauscript is orgaizd as follows: W dscrib t systm udr ivstigatio, composd of a optomcaical systm ad a additioal rfrc driv i Sc. II. Sctio III cotais t aalytical drivatio of t microscopic Adlr quatios. A major part of tis drivatio is do by applyig t lasr tory for optomcaical limit cycls 30] to tis problm. Tis is prstd i t Appdix. I Sc. IV w dmostrat umrically tat sycroizatio is xpctd also i a quatum paramtr rgim.

2 FIG. 1. Scmatics of a gric optomcaical systm. I a rotatig fram wit frqucy ω L, obtaid by applyig t uitary trasformatio Û = xp ( iω L ta a ), t cavity wit frqucy is driv by a tim-idpdt lasr, dpictd by t black arrow to t lft of t cavity. I t slf-oscillatory rgim of t mcaical oscillator wit atural frqucy ω m, t mcaical oscillator may sycroiz to a additioal optical driv wit frqucy ω op as dpictd i dasd box (a), or to a mcaical driv wit frqucy ω m as dpictd i dasd box (b). Not tat ω op is giv i t rotatig fram, wil t applicatio of Û lavs bot ω m ad ω m idtical i bot frams. II. THE SYSTEM T stadard Hamiltoia of a optomcaical systm i wic t positio of t mcaical oscillator paramtrically modulats t frqucy of a lctromagtic cavity is giv i a fram rotatig wit t frqucy of t lasr driv, ω L, by ] H = ω m b b a a g 0 a a ( b + b ) ia L ( a a ), (1) wr a ad a ar t cratio ad aiilatio oprators of potos i t cavity, b ad b ar t cratio ad aiilatio oprators of poos i t mcaical rsoator, ω m is t mcaical frqucy of oscillatio, = ω L ω c is t dtuig from cavity rsoac at ω c of t drivig fild wit strgt A L, g 0 is t sigl poto couplig costat, ad w av st = 1. A scmatic figur of t systm is sow i Fig. 1. T fram rotatig wit lasr driv ω L is obtaid by applyig t uitary trasformatio Û = xp ( iω L ta a ), wic grats t Hamiltoia ÛH old Û iû Û / t. T dissipatio of t two oscillators (t mcaical rsoator ad t optical cavity) ca b dscribd via t mastr quatio, wit t Lidblad oprators ad dρ dt = i H, ρ ] + L m ρ + L c ρ, () L m ρ = γ m ( t + 1)Db]ρ + γ m t Db ]ρ, (3) L c ρ = γ c Da]ρ, (4) FIG.. Wigr fuctio rprstatio of (a) slf-sustaid oscillatios i t mcaical oscillator ad of (b) tdcy towards pas-lockig of t mcaical oscillator to t pas of a sycroizig rfrc driv. T paramtrs usd for bot plots ar (g 0, γ c, γ m, A L,, t ) = (0.3, 0.3, 0.015, 0.4, 0, 0) ω m, wr t paramtrs of t xtral optical driv i (b) ar (A op, ω op ) = (0.08, 0.98) ω m. wr γ m ad γ c ar t amplitud dampig rats of t mcaical oscillator ad t lctromagtic cavity corrspodigly, t is t ma poo umbr i trmal quilibrium, ad Dx]ρ = xρx ( ρx x + x xρ ) /. I tis work w would lik to study t sycroizatio of t mcaical lmt of t optomcaical systm to a xtral driv. W cosidr two cass: Cas (1) - Optical lasr driv - W itroduc a additioal optical lasr rfrc fild, wit frqucy ω op giv i a fram rotatig wit frqucy ω L, ad strgt A op, by addig t trm ( H op = ia op iω op t a iωop t a ) (5) to t Hamiltoia apparig i Eq. (). Tis is dpictd i dasd box (a) i Fig. 1. Tis Hamiltoia ca b ralizd by a additioal optical lasr, or, if t mcaical frqucy ω m is larg oug suc tat a rotatig-wav approximatio ca b usd, by priodically modulatig t powr of t optical lasr causig t mcaical slf-oscillatios, as s i Rf. 35]. Cas () - Mcaical driv - A mcaical rfrc driv wit frqucy ω m ad strgt A m ca b applid dirctly oto t mcaical oscillator,.g., by itroducig a pizolctric compot as dpictd i dasd box (b) i Fig. 1. I aalogy to cas (1), w add t trm ( H m = ia m iω m t b iωm t b ) (6) to t Hamiltoia apparig i Eq. (). Slf-oscillatios i t optomcaical systm.- A optomcaical systm driv by a ffctiv blu-dtud lasr may giv ris to slf-sustaid oscillatios i t mcaical oscillator 7 3]. Ts slf-oscillatios ar a prrquisit for studyig sycroizatio. Ty ca b illustratd i pas-spac via t Wigr fuctio pas-spac distributio. A Wigr fuctio rprstatio for a spcific slf-sustaid oscillatio i t mcaical oscillator is sow i Fig. (a). Udr t ifluc of a rfrc driv, t mcaical slf-oscillator may dvlop a pas-prfrc as it tds towards lockig oto t pas of t driv. For a additioal

3 3 optical lasr driv, as i cas (1), t Wigr rprstatio for a mcaical slf-oscillator sowig suc pas-prfrc is sow i Fig. (b)38]. III. SYNCHRONIZATION - ANALYTICAL CALCULATION I t followig sctio, it is our goal to driv a aalytical dscriptio for t sycroizatio of t mcaical slfoscillator to a rfrc driv. To do so, w apply t lasr tory for optomcaical limit cycls 30] to t currt problm, Eq. (), i wic a additioal rfrc driv, Eq. (5) or Eq. (6), is iflucig t optomcaical limit cycl. Tis approac is basd o t assumptio tat t dyamics of t cavity adiabatically follows t dyamics of t mcaical oscillator. It allows us to obtai a quatio of motio (EOM) for t pas distributio of t slf-oscillator, σ(r, φ), wr r ad φ ar t mcaical pas-spac variabls rprstig t radius ad t pas of t slf-oscillator. To kp tis mauscript cort ad focusd o sycroizatio, w sift t drivatio of t rlvat EOMs to t Appdix. Hr i t mai txt, w will us t drivd EOMs as a startig poit. Cas (1) - Optical lasr driv - As xplaid i t Appdix, t EOM for σ(r, φ) is valid w t dyamics of t cavity adiabatically follows t dyamics of t mcaical oscillator, t optomcaical couplig is small, g 0 ω m, ad t trmal- ad quatum sot-ois dos ot play a rol. I a rotatig fram wit frqucy ω m, t EOM for σ(r, φ) is of t form σ = ( r µ r + φ µ φ ) σ, (7) wr t pas-drift cofficit is giv by, µ φ = 1 r ] J J g 0 A L {A L R = +A op iφ ] J J iɛt + A op iφ J J iɛt ]}, (8) ad t xplicit xprssios for t radius-drift cofficit is giv i Eq. (A18). I t last xprssio J := J ( g 0 r/ω m ) is t Bssl fuctio of t first kid of t t ordr, ɛ ω m is t dtuig btw t frqucis of t rfrc driv ad t atural frqucy of t mcaical oscillator, ad is dfid as ω op = γ c + i(ω m ff ), (9) wr t dfiitio for t ffctiv dtuig of t cavity fild, ff, is giv i Eq. (A10). Tis EOM dscribs t dyamics of t mcaical oscillator ad, i a appropriat paramtr rgim, will trfor dscrib t sycroizatio of t mcaical oscillator oto t optical rfrc driv. T ost of sycroizatio is caractrizd by t lockig of t pas of t mcaical oscillator to t pas of t optical driv, wil t radius of oscillatio stays approximatly costat. For tat raso, w ca glct t trm dscribig t drift of t radius, µ r, wil focusig o t drift of t pas, Eq. (8). W ar trfor lft wit σ = φ µ φ σ, (10) from wic w rcogiz tat µ φ = φ. Lt us trfor focus o µ φ, Eq. (8), wic compltly dtrmis t tim volutio of φ. T first trm is t kow amplitud-dpdt optomcaical frqucy sift δω (sf. 3]), i.., w obtai µ φ = φ = δω + g 0A L A op i(φ+ɛt) J J + i(φ+ɛt) ] J J R r. (11) I t sidbad-rsolvd rgim ad wit a dtuig clos to t mcaical frqucy, i.., γ c / ff ω m, trms wit 1 i t domiator ar clos to rsoac. For tat raso, w will kp oly t trms wit = 1,. W t fid φ = δω + A op,ff si(φ + ɛt), (1) wr w av siftd φ by a costat ad dfid t ffctiv driv strgt as A op,ff = g 0 A L A op ( ) 1 + γ c 4ω m rω m (J + J 0 ) J0 + 4ω m (J γc J 0 J1 J 0 ). (13) Addig t frqucy diffrc ɛ to bot sids of Eq. (1), w obtai t Adlr quatio δφ = (ω op ω ff m ) + A op,ff si(δφ), (14) wr t ffctiv mcaical frqucy is ω ff m ω m + δω, ad w av dfid δφ φ + ɛt. Not tat δφ = (φ ω m t) + ω op t is just t diffrc of pas of t mcaical oscillator (i a fram rotatig wit ω m ) to t pas of t xtral driv. T Adlr quatio dscribs t sycroizatio of t mcaical slf-oscillator to t rfrc driv, as sow i Fig. 3, i wic w plot si δφ as a fuctio of (ω op ω ff m ) for diffrt driv strgts, wr t ovrli rfrs to timavragig. For ω op ω ff m < A op,ff, t solutio to Eq. (14) is δφ = 0. Trfor pas-lockig taks plac. For ω op ω ff m A op,ff, si(δφ) tim-avrags to zro. T optomcaical paramtrs cos i Fig. 3 ca b radily obtaid i a wid rag of xprimts, 4, 39]. I Rf. 39] a mcaical rsoator of frqucy ω m /(π) = 9.7(kHz) was studid, wil i Rf. 4] a mcaical rsoator of frqucy ω m /(π) = 3.9(GHz) was studid. I bot, t paramtrs of t optomcaical systm wr similar to tos giv i Fig. 3.

4 4 FIG. 3. Sycroizatio of t mcaical slf-oscillator to a optical rfrc driv. T aalytically calculatd tim-avrag op ff si(δφ) as a fuctio of (ωop ωm ) for diffrt valus of A, from 0.13 to T ist compars t aalytical solutio (blu) wit t umrical simulatio (rd dasd) for Aop = It sows xcllt agrmt. Colord rgio idicats t sycroizatio rgio, dδφ/dt = 0. T paramtrs of t optomcaical systm ar tak i t classical rgim, (g0, γc, γm, AL,, t ) = (0.015, 0.5, , 1.0, 1.0, 0) ωm. FIG. 4. Sycroizatio of t mcaical slf-oscillator to a mcaical rfrc driv. T aalytically calculatd tim-avrag m si(δφ) as a fuctio of (ωm ωff m ) for diffrt valus of A, from to T ist compars t aalytical solutio (blu) wit t umrical simulatio (rd dasd) for Am = It sows xcllt agrmt. Colord rgio idicats t sycroizatio rgio, dδφ/dt = 0. T paramtrs of t optomcaical systm ar tak i t classical rgim, (g0, γc, γm, AL,, t ) = (0.01, 0.3, , 1.0, 1.0, 0) ωm. W ca furtr tst tis drivd Adlr quatio by comparig it wit t umrical prdictio, wic ca b obtaid by itgratig t optomcaical quatios of motio for t cavity fild α ad t mcaical fild β 3] Tis form of t Adlr quatio agrs wit 33]. I Fig. 4 ff w plot si δφ as a fuctio of (ωop ωm ) for diffrt driv strgts, wr t ovrli rfrs to tim-avragig. W ca furtr tst tis aalytical quatio by comparig it wit t classical umrical prdictio, wic ca b obtaid by itgratig t quatios of motio α = i α + ig0 (β + β )α β = ig0 α iωm β γc iωop t α + AL + Aop, γm β. (15) (16) T rsult is sow i t ist of Fig. 3. T sycroizatio rgio is idicatd by t colord rgio. Tr is a vry good agrmt btw t prdictio of t drivd microscopic quatio ad t umrical simulatio. Cas () - Mcaical driv - Aalogously to cas (1), by applyig t lasr tory for optomcaical limit cycls w obtai a EOM for t pas distributio of t slf-oscillator, σ(r, φ). Tis EOM as t sam form as Eq. (7), but wit a pas-drift cofficit wic is giv by # " m 1 J J X m A si (ω ωm )t + φ, µφ = g0 AL R r (17) ad wit a radius-drift cofficit wic is giv i t Appdix, Eq. (A0). Now, takig idtical stps to tos sow i cas (1), o racs a Adlr quatio, ff m,ff = (ωm δφ si(δφ), ωm ) + A (18) wr t ffctiv driv strgt is Am,ff = Am. r (19) α = i α + ig0 (β + β )α β = ig0 α iωm β γc α + AL, γm iωm t. β + Am (0) (1) T compariso is s i t ist of Fig. 4. A vry good agrmt is foud btw t aalytical Adlr quatio ad t umrical simulatio. IV. QUANTUM SYNCHRONIZATION NUMERICAL DEMONSTRATION T optomcaical systm is tortically suggstd to dmostrat sycroizatio also i a quatum paramtr rgim, i wic g0 ωm dos ot old aymor. I tat paramtr rgim, t quatum sot ois plays a importat rol, ad caot b glctd as i t prvious sctio. T quatum sycroizatio of two suc systms was tortically studid i Rf. 18]. Hr w sow umrically tat t mcaical slf-oscillator is xpctd to sycroiz to a rfrc driv i t quatum paramtr rgim. Bfor discussig t umrical calculatio ad t rsults, w itroduc t sycroizatio masur usd.

5 5 FIG. 5. T tim-avragd sycroizatio masur S as a fuctio of t xtral driv s frqucy, sow i blu for a optical driv wit A op /ω m = 0.08 ad i a rd dasd li for a mcaical driv wit A m /ω m = For t mcaical driv tr is oly o sycroizatio pak at ω m = ω ff m, wil t optical driv lads to multipl sycroizatio paks at ω op = { } ω ff m /3, ω ff m /, ω ff m, ω ff m. T black dottd lis ar plottd at ts frqucis. T sycroizatio paks at ω op = { } ω ff m /3, ω ff m ar ardly oticabl i t scal of t figur, ad ar trfor sow i t two ists. Optomcaical paramtrs ar t sam as i Fig.. A. T sycroizatio masur Sycroizatio of a slf-oscillator to a xtral driv is t dvlopmt of pas prfrc for t slf-oscillator as it tds towards pas-lockig to t pas of t xtral driv. As sow i Fig., tis pas prfrc is asily s i t pas-spac distributio of t mcaical oscillator. T iformatio stord i t pas-spac distributio ca b accoutd for by usig a sigl umbr 8], S = b b b, () wr t brackt... dots avragig ovr t passpac distributio. T umrator olds iformatio rgardig t sprad of t pas-spac distributio, wil t domiator is itroducd for t purpos of ormalizatio. For a compltly pas-idpdt limit cycl ctrd aroud zro, t oscillator is obviously compltly usycroizd, ad idd w will fid S = 0. As t slf-oscillator sycroizs to a xtral driv, a pas prfrc dvlops. Tis will rduc t pas variatio, rsultig i largr valus of S. For a cort stat t masur is S = 1, maig t oscillator is strogly sycroizd to t driv. Tis masur caot b usd i cass for wic t slf-oscillator dvlops multipl prfrrd pass. Not tat i t optomcaical systm, t slfoscillatios dvlopig i t mcaical oscillator ar ctrd aroud som poit i pas-spac, β c, wic is grally diffrt ta t origi. Tis is s i Fig. (a). Tis dviatio from t origi iflucs t sycroizatio masur, Eq. (). Tis ca b asily corrctd ad accoutd for. To do so, w mov to a displacd fram by usig t displacmt oprator D( β c ) = xp( β c b + β c b). For t rst of tis work, w will b workig i t appropriat displacd fram. T problm of a optomcaical systm wit a additioal rfrc driv, Eq. () wit itr Eq. (5) or Eq. (6), cotais a tim-dpdt Hamiltoia. For tat raso, a stady stat dos ot mrg. Howvr, i t lat-tim dyamics, t systm volvs ito a stat wic is priodic i tim wit priodicity τ π/ω i, wr i dots t opticalor t mcaical-rfrc driv. Tis is tru i t sycroizd stat ad outsid t sycroizd stat, ad it is t rsult of t priodic tim dpdc of t Hamiltoia. For tat raso, i t lat-tim dyamics t sycroizatio masur S is a fuctio of tim wit t sam priodicity, S (t) = S (t + τ). T variatio of S ovr t tim scal τ i t lat-tim dyamics is rlativly small, ad is of ordr S 0.01 at maximum. To covitly discuss sycroizatio, w us S, dfid as t tim-avrag of S ovr a priod τ. B. Numrical Rsults To umrically study sycroizatio of t mcaical slf-oscillator to a xtral driv, w us QuTiP 40, 41]. Cas (1) - Optical lasr driv - I Fig. 5, t tim-avragd sycroizatio masur S is plottd i blu as a fuctio of t frqucy of t rfrc driv, ω op. A mai sycroizatio pak appars about a ffctiv mcaical frqucy, ω ff m, sligtly siftd from ω m. Tis sift of t mcaical frqucy is kow 31, 3] to b t rsult of t avrag dyamics of t lctromagtic cavity. Sycroizatio paks at otr frqucis ar foud as wll: A sycroizatio pak about ω op = ω ff m / is clarly visibl, ad i t ists of Fig. 5 w zoom i o t vry small sycroizatio paks at ω op = { } ω ff m /3, ω ff m. Ts sycroizatio paks ar kow i t litratur as ig-ordr sycroizatio 1, ]. Wil i pricipl ig-ordr sycroizatio is always prst w sycroizig a slf-oscillator to a rfrc driv, it is i practic vry difficult (if ot impossibl) to dtct. T prsc of a rfrc driv wic cotais may frqucy compots i its oscillatio ca ac t sycroizatio paks ]. As sow i t Appdix, t ffctiv driv of t mcaical slf-oscillator, Eq. (A6), idd cotais multipl frqucis. For tat raso, ad i cotrast to cas (), w ca obsrv t smallr sycroizatio paks. W ca also otic a asymmtry i t sycroizatio pak wit rspct to t rfrc fild s frqucy. Tis ca b also b s i Fig. 6. Wil tr is o raso to xpct prfct symmtry, it is visibl tat t cas of a optical rfrc driv is mor asymmtric. Tis is du to t ig-ordr sycroizatio paks. ω ff I Fig. 6 (a) w focus o t sycroizatio pak for ω op = m. Tis corrspods to t maximal sycroizatio pak sow i Fig. 5. T sycroizatio masur S is plottd as a fuctio of bot A op ad ω op. Idd, t Arold togu

6 microscopic mastr quatio, w aalytically dvlop Adlr quatios dscribig t sycroizatio of t mcaical slf-oscillator to a rfrc driv. Tis was do for two diffrt rfrc drivs, a optical o ad a mcaical o (as was sow i Rf. 33]). W also sow umrically tat sycroizatio i a quatum paramtr rgim is xpctd, trfor suggstig t optomcaical systm as a good cadidat for t study of quatum sycroizatio. 6 ACKNOWLEDGMENTS Tis work was fiacially supportd by t Swiss SNF ad t NCCR Quatum Scic ad Tcology. A.N. olds a Uivrsity Rsarc Fllowsip from t Royal Socity ad ackowldgs support from t Wito Programm for t Pysics of Sustaiability. Appdix A: Applyig t Lasr Tory for Optomcaical Limit Cycls FIG. 6. Arold Togu: T sycroizatio masur S is plottd as a fuctio of t driv frqucy ad strgt for cas (1) (a), ad for cas () (b). S as t typical sap of a Arold togu. T black lis mark ω op = ω ff m ad ω m = ω ff m i (a) ad (b) corrspodigly. T orizotal wit lis mark t cut alog wic Fig. 5 is plottd. Optomcaical paramtrs ar t sam as i Fig.. is prst, a sigatur for sycroizatio. Cas () - Mcaical driv - T rfrc driv sycroizs t mcaical oscillator at frqucy ω m = ω ff m. Tis is sow by t rd dasd curv i Fig. 5. I cotrast to t optical cas, o ig-ordr sycroizatio is s. Idd, as t mcaical driv is actig dirctly o t mcaical slf-oscillator, its ifluc is armoic. Trfor ig-ordr sycroizatio is ot dtctd 1, ]. I Fig. 6 (b) w focus o tis sycroizatio pak. I tis figur w vary bot t xtral frqucy ω m ad t strgt of t xtral driv, A m, ad t Arold togu is clarly obsrvd. V. CONCLUSIONS I coclusio, our work fills a gap i t study of sycroizatio of a optomcaical systm. Startig from t I t followig, w apply lasr tory for optomcaical limit cycls 30] to our currt problm, Eq. (), wit a additioal rfrc driv as giv by Eq. (5) or Eq. (6). W driv a EOM for σ, t pas-spac distributio of t mcaical oscillator, for ac cas. I applyig lasr tory to our problm, t iitial stps ar idtical to ts tak w applyig lasr tory to a bar optomcaical systm (wit o rfrc driv). W trfor do ot rpat ts stps, but rfr to Rf. 30] for our startig poit, Eq. (A), wic is prstd blow. A sort summary of t stps tak i tis Appdix: Switcig to a pas-spac rprstatio for t mcaical oscillator dgr of frdom. Tis allows us to us a diffrt adiabatic rfrc stat of t lctromagtic cavity fild for ac poit i t pas-spac of t mcaical oscillator. Assumig t lctromagtic cavity dyamics follows adiabatically t dyamics of t mcaical oscillator, w solv for a approximat solutio for t cavity fild. W us t solutio for t cavity fild as a rfrc stat for t mcaical stat, wic allows us to obtai a EOM for σ. Cas (1) - Optical lasr driv - T mastr quatio dscribig our systm, Eq. () wit a optical rfrc driv, Eq. (5), ca b writt i a pas-spac pictur for t mcaical oscillator 30, 4, 43]. T systm is t dscribd by σ(β, β ), wic is a dsity oprator for t lctromagtic cavity fild ad a quasi-probability distributio for t mcaical oscillator, wit β rprstig t mcaical pas-spac variabl. Tis rsults i a dpdc of t cavity dtuig o t pas-spac variabls of t mca-

7 7 ical oscillator. If w furtr us t smipolaro trasformatio 30], σ(β, β, t) = g 0(β β )a a/ω m σ(β, β, t) g 0(β β )a a/ω m, (A1) tis dpdc is covitly trasformd ito o of t drivig fild. T trasformd mastr quatio, i a mcaical fram rotatig wit frqucy ω m, is t t σ(β, β, t) = (L m + L c + L it ) σ(β, β, t), (A) wit L m σ = γ ( m β β + β β ) σ + γ m ( t + 1/) β βσ, (A3) L c σ = i a a K(a a) i ( E(t) a E(t)a ), σ ] (A4) + γ c Da]σ, L it σ = i g ( ) 0 iω m t β iωmt β σa a + H.c. (A5) wr t Krr oliarity is K = g 0 /ω m, ad E is a gralizd drivig fild wic dpds o t mcaical stat, E(t) = ( A L + A op ) iωop t = J ( ηr) i(ω mt φ). (A6) Hr, J is t Bssl fuctio of t first kid of t t ordr, r ad φ ar t mcaical pas-spac variabls β = r iφ, ad η = g 0 /ω m. W will us t sortad otatio J := J ( ηr). W will ow assum tat t cavity dyamics, wit a domiat tim scal γc 1, is fast compard to all otr tim scals i L m ad L it. Tis mas tat w ca solv for t cavity fild α(t), wil assumig t stat of t mcaical oscillator, dscribd by t pas-spac variabls β ad β, is fixd. T solutio for α(t) will t b cosidrd as a classical rfrc amplitud. 1. Calculatig t classical rfrc amplitud Udr t assumptio tat t cavity dyamics wit caractristic tim scal γc 1 is muc fastr ta all otr dyamics i t problm, its stat is govrd by L c, wil t ffct of L it is glctd, σ = L c σ. (A7) Tis quatio dscribs a cavity wit Krr oliarity, wic is driv by two amplitud ad pas modulatd filds. A approximat solutio to Eq. (A7) ca b foud i two limits: t limit α(t) 1, i.., a cavity wic is driv by a strog driv to a stat of larg ma amplitud, ad t limit α(t) 1, for wic t cavity is driv by a wak driv ad stays clos to its groud stat. I t formr limit w glct trms up to first ordr i α, wil i t lattr w glct trms of tird ordr i α: a. α(t) 1 From Eq. (A7), o ca obtai a EOM for t classical cavity fild amplitud α(t), { γc α(t) = i + K α(t) ] } α(t) + E(r, φ, t). (A8) As a rsult of t form of t drivig fild E, Eq. (A6), w xpct t solutio to b of t form α(r, φ, t) = = α l (r, φ) iω mt + α (r, φ) i(ω m+ω op )t ], (A9) wr t amplituds α l (r, φ) ad α (r, φ) sall b dtrmid. As oticabl from Eq. (A8), t ffctiv dtuig flt by t lctromagtic cavity dpds o α(t). By placig t solutio obtaid for α(t), wil kpig oly t domiat dc compots, w obtai for t ffctiv dtuig ] ff (r, φ) = +K α l + α + α l (α ) + (α l +1 ) α, (A10) wr w av assumd ω op = ω m + ɛ, wit ɛ ω op, ω m. Tis assumptio is satisfid w studyig sycroizatio i t viciity of ω op ω m. W solv Eq. (A8) by assumig a fixd ffctiv dtuig ff. W t fid, wr w av usd ω op foud a solutio for α(t). α l = A L J ( ηr) iφ, α = Aop J ( ηr) iφ, +1 (A11) (A1) = γ c + i(ω m ff ), (A13) = ω m + ɛ agai. W av trfor b. Displacd fram for α(t) 1 I tis limit, Eq. (A7) dictats tat α(t) sould solv α(t) = i( + K) γ ] c α(t) + E(r, φ, t). (A14) As compard wit Eq. (A8), w s tat a diffrt ffctiv dtuig sould b dfid, K = + K. T, w ca procd as was do i t α(t) 1 limit. Rsults will b i complt aalogy, ad ca b obtaid by cagig ff K.. Obtaiig t EOM Aftr fidig t solutio for α(t), wic will srv as a classical rfrc amplitud for t mcaical oscillator, our xt goal is to obtai t EOM for t pas-spac distributio of t mcaical oscillator, σ(β, β ). W otic tat

8 8 t dyamics of tis pas-spac distributio ar govrd by Eq. (A3) ad Eq. (A5). By placig t solutio for α(t) ito Eq. (A5), o obtais a EOM for t pas-spac distributio of t mcaical oscillator, σ = ig 0 β α l (α l ) + α l (α ) iɛt + α (α l ) iɛt] σ + H.c., (A15) wr w av glctd trms proportioal to (A op ), kpt oly dc trms, ad av usd ω op = ω m + ɛ, wr ω op, ω m ɛ. Tis allowd us to sd +1, wil kpig t xpotials dpdig o ɛ, as ty will b dd latr to dscrib sycroizatio. I dscribig limit cycls ad sycroizatio, it is mor atural to work i polar coordiats. W trfor trasform Eq. (A15) to a polar coordiat systm. A mor dtaild dscriptio of tis trasformatio ca b foud i Rf. 30]. T trasformd EOM is t, σ = r µ r φ µ φ ] σ, wr t drift cofficits ar giv by (A16) µ φ = 1 ] J J g 0 A L {A L R r +A op iφ ] J J iɛt + A op iφ ]} J J iɛt, (A17) µ r = γ ] m r + J J g 0 A L {A L Im +A op iφ ] J J Im iɛt + A op iφ ]} J J Im iɛt, (A18) wr w av glctd trms 1/r i t quatio for µ r ad trms 1/r i t quatio for µ φ, ad av icludd t ffct du to Eq. (A3). I t limit of A op 0, o rtrivs t kow xprssio from 30]. Cas () - Mcaical driv - I applyig t lasr tory for optomcaical limit cycls for tis cas, w tak stps compltly aalogous to tos tak i t prvious cas. As t mcaical rfrc driv acts dirctly o t mcaical slf-oscillator, it dos ot appar i t solutio for α(t) or i t limiatio of t lctromagtic cavity. Tis fact maks calculatios mor straigtforward i t prst cas, ad w do ot xplicitly prst tm r. T EOM obtaid as t sam form as Eq. (A16), wit drift cofficits wic ar giv by µ φ = 1 ] g r 0 A L R J J A m si (ω m ω m )t + φ ], (A19) µ r = γ ] m r + g 0 A L Im J J + A m cos (ω m ω m )t + φ ]. (A0) As i t prvious cas, w av glctd trms 1/r i t quatio for µ r ad trms 1/r i t quatio for µ φ. I t limit of A m 0, o rtrivs t kow xprssios from 30]. Appdix B: Fokkr-Plack Equatio for t Mcaical Slf-Oscillator Usig lasr tory for optomcaical systms allows o to obtai a Fokkr-Plack quatio (FPE) dscribig t dyamics of t mcaical slf-oscillator. Tis FPE is of t form Ẇ = r µ r φ µ φ + rrd rr + rφd rφ + φφd φφ ] W, (B1) wr W(r, φ) is cos to b t Wigr pas-spac distributio, µ r ad µ φ ar t drift cofficits of t pas-spac variabls r ad φ corrspodigly, ad D rr, D rφ ad D φφ ar t diffusio cofficits. I App. A w aimd to obtai oly t drift cofficits, as ty ar sufficit to dscrib sycroizatio i a paramtr rgim i wic t diffusio dos ot play a sigificat rol. For compltio ad for tos itrstd, w giv i tis appdix t xprssios for t diffusio cofficits. Cas (1) - Optical lasr driv - T drift cofficits of t FPE quatio ar giv i Eqs. (A17)-(A18), wil t diffusio cofficits ar giv by

9 9 D φφ = 1 r γ m ( t + 1/) 4 D rφ = 1 r γ c g 0 A L +1 D rr = γ m( t + 1/) γ c g 0 A L r 8 +1 A J L + A J ] + L + + A J+ J L R + A op J + (J +1 + J ) cos(φ + ɛt) +A op iφ ] J + J iɛt + A op 3iφ ]} J J +1 + iɛt, + { ] J+ J A L Im + A op iφ ] J + J Im + iɛt + A op 3iφ J J +1 Im (B) iɛt ]}, (B3) γ c g 0 A L 8 +1 A J L + A J ] + L + A J+ J L R + Aop J (J J ) cos(ɛ + δt) A op iφ ] J + J iɛt A op 3iφ J J iɛt ]}. (B4) Cas () - Mcaical driv - T drift cofficits of t FPE quatio ar giv i Eqs. (A19)-(A0), wil t rfrc fild A m dos ot tr t xprssios for t diffusio. T diffusio cofficits ar trfor giv i Eqs. (B)-(B4), wit A op = 0. 1] A. Pikovsky, M. Rosblum, ad J. Kurts, Sycroizatio: A Uivrsal Cocpt i Noliar Scics (Cambridg Uivrsity Prss, 003). ] A. Balaov, N. Jaso, D. Postov, ad O. Sosovtsva, Sycroizatio: From Simpl to Complx (Sprigr, Nw York, 008). 3] S. H. Strogatz, Noliar Dyamics ad Caos: Wit Applicatios to Pysics, Biology, Cmistry, ad Egirig (Wstviw Prss, 001), d d. 4] S. Waltr, A. Nukamp, ad C. Brudr, Pys. Rv. Ltt. 11, (014). 5] S. Waltr, A. Nukamp, ad C. Brudr, A. Pys. 57, 131 (015). 6] T. E. L ad H. R. Sadgpour, Pys. Rv. Ltt. 111, (013). 7] T. E. L, C.-K. Ca, ad S. Wag, Pys. Rv. E 89, 0913 (014). 8] N. Lörc, E. Amitai, A. Nukamp, ad C. Brudr, Pys. Rv. Ltt. 117, (016). 9] T. Wiss, S. Waltr, ad F. Marquardt, Pys. Rv. A 95, (017). 10] V. M. Bastidas, I. Omlcko, A. Zakarova, E. Scöll, ad T. Brads, Pys. Rv. E 9, 0694 (015). 11] V. Amri, M. Egbali-Arai, A. Mari, A. Farac, F. Kiradis, V. Giovatti, ad R. Fazio, Pys. Rv. A 91, (015). 1] N. Lörc, S. E. Nigg, A. Nukamp, R. P. Tiwari, ad C. Brudr, -prit arxiv: (017). 13] C. Davis-Tilly ad A. D. Armour, Pys. Rv. A 94, (016). 14] M. Xu, D. A. Tiri, E. C. Fi, J. K. Tompso, ad M. J. Hollad, Pys. Rv. Ltt. 113, (014). 15] A. Rot ad K. Hammrr, Pys. Rv. A 94, (016). 16] B. Zu, J. Scacmayr, M. Xu, F. Hrrra, J. G. Rstrpo, M. J. Hollad, ad A. M. Ry, Nw. J. Pys. 17, (015). 17] M. R. Hus, W. Li, S. Gway, I. Lsaovsky, ad A. D. Armour, Pys. Rv. A 91, (015). 18] T. Wiss, A. Krowald, ad F. Marquardt, Nw. J. Pys. 18, (016). 19] M. Ludwig ad F. Marquardt, Pys. Rv. Ltt. 111, (013). 0] X.-F. Yi, W.-Z. Zag, ad L. Zou, J. Mod. Opt. 64, 578 (017). 1] G. L. Giorgi, F. Galv, G. Mazao, P. Colt, ad R. Zambrii, Pys. Rv. A 85, (01). ] M. Asplmyr, T. J. Kippbrg, ad F. Marquardt, Rv. Mod. Pys. 86, 1391 (014). 3] J. D. Tufl, T. Dor, D. Li, J. W. Harlow, M. S. Allma, K. Cicak, A. J. Sirois, J. D. Wittakr, K. W. Lrt, ad R. W. Simmods, Natur 475, 359 (011). 4] J. Ca, T. P. M. Algr, A. H. Safavi-Naii, J. T. Hill, A. Kraus, S. Groblacr, M. Asplmyr, ad O. Paitr, Natur 478, 89 (011). 5] T. A. Palomaki, J. W. Harlow, J. D. Tufl, R. W. Simmods, ad K. W. Lrt, Natur 495, 10 (013). 6] T. P. Purdy, R. W. Ptrso, ad C. A. Rgal, Scic 339, 801 (013). 7] F. Marquardt, J. G. E. Harris, ad S. M. Girvi, Pys. Rv. Ltt. 96, (006). 8] C. Mtzgr, M. Ludwig, C. Nua, A. Ortlib, I. Favro, K. Karrai, ad F. Marquardt, Pys. Rv. Ltt. 101, (008). 9] J. Qia, A. A. Clrk, K. Hammrr, ad F. Marquardt, Pys. Rv. Ltt. 109, (01). 30] N. Lörc, J. Qia, A. Clrk, F. Marquardt, ad K. Hammrr, Pys. Rv. X 4, (014). 31] D. A. Rodrigus ad A. D. Armour, Pys. Rv. Ltt. 104, (010). 3] A. D. Armour ad D. A. Rodrigus, C. R. Pys. 13, 440 (01). 33] G. Hiric, M. Ludwig, J. Qia, B. Kubala, ad F. Marquardt, Pys. Rv. Ltt. 107, (011). 34] C. A. Holms, C. P. May, ad G. J. Milbur, Pys. Rv. E 85, (01). 35] K. Slomi, D. Yuvaraj, I. Baski, O. Sucoi, R. Wiik, ad

10 10 E. Buks, Pys. Rv. E 91, (015). 36] M. Zag, G. S. Widrckr, S. Maipatrui, A. Barard, P. McEu, ad M. Lipso, Pys. Rv. Ltt. 109, (01). 37] M. Zag, S. Sa, J. Cardas, ad M. Lipso, Pys. Rv. Ltt. 115, (015). 38] Not1, t ifluc of t rfrc driv, idally, sould ot cag t amplitud of t slf-oscillator. I practic owvr, tr is som ifluc. I tis work w mak sur tat t rfrc driv dos ot cag t amplitud of t usycroizd limit cycl by mor ta 10%. 39] D. Klckr, B. Pppr, E. Jffry, P. Soi, S. M. To, ad D. Bouwmstr, Opt. Exprss 19, (011). 40] J. Joasso, P. Natio, ad F. Nori, Comput. Pys. Commu. 183, 1760 (01). 41] J. Joasso, P. Natio, ad F. Nori, Comput. Pys. Commu. 184, 134 (013). 4] F. Haak ad M. Lwsti, Pys. Rv. A 7, 1013 (1983). 43] C. Gardir ad P. Zollr, Quatum Nois (Sprigr-Vrlag Brli Hidlbrg, 004).