Stochastic Phase Space Localization for a Single Particle

foraSingleTrappedParticle

StefanoMancini∗,DavidVitaliandPaoloTombesi

DipartimentodiMatematicaeFisica,Universit`adiCamerino,viaMadonnadelleCarceriI-62032Camerino

andIstitutoNazionaleperlaFisicadellaMateria,Camerino,Italy

(February1,2008)

Weproposeafeedbackschemetocontrolthevibrationalmotionofasingletrappedparticlebasedonindirectmeasurementsofitsposition.Itresultsthepossibilityofamotionalphasespaceuncertaintycontraction,correspondingtocooltheparticleclosetothemotionalgroundstate.

arXiv:quant-ph/9810022v5 14 Feb 2000I.INTRODUCTION

Inrecentyearstherehasbeenanincreasinginterestontrappingphenomenaandrelatedcoolingtechniques[1].Someyearsagoithasbeenshownthat,usingresolvedsidebandcooling,asingleioncanbetrappedandcooleddownneartoitszero-pointvibrationalenergystate[2]andrecently,analogousresultshavebeenobtainedforneutralatomsinopticallattices[3].Thepossibilitytocontroltrappedparticles,indeed,gaverisetonewmodelsinquantumcomputation[4],inwhichinformationisencodedintwointernalelectronicstatesoftheionsandthetwolowestFockstatesofavibrationalcollectivemodeareusedtotransferandmanipulatequantuminformationbetweenthem.Itmayhappenhowever,thatatrappedionthatisafavorablecandidateforquantuminformationprocessingsinceitpossesesahyperfinestructurewithlongcoherencetimes(asforexample25Mg+[5]),isnotsuitableforresolvedsidebandcooling.Insuchcasesitmaybehelpfultohaveanalternativecoolingtechnique,whichcanbeappliedwhenresolvedsidebandcoolingisimpracticaltouse.

Inthispaperwepresentawaytocontrolthemotionofatrappedparticle,whichisabletogiveasignificantphase-space-localisation.Thebasicideaoftheschemeistorealizeaneffectiveandcontinuousmeasurementofthepositionofthetrappedparticleandthenapplyafeedbackloopabletodecreasethepositionfluctuations.Duetothecontinuousnatureofthemeasurementandtotheeffectofthetrappingpotentialcouplingtheparticlepositionwithitsmomentum,feedbackwillrealizeaneffectivephase-spacelocalisation.

Withthisrespecttherearesomeanalogiesbetweenthepresentmethodandresolved-sidebandstimulatedRamancooling[6],whichcanbeviewedasasortoffeedbackscheme.Infact,oneofthetwoRamanlasersperformsaneffectivemeasurementofthevibrationalnumberbychangingtheparticleinternalstateonlyifitisanexcitedvibrationalstate.ThesecondRamanlaserperformsinsteadthefeedbackstep,becauseitputstheparticlebackintheinitialinternalstate,afterhavingremovedavibrationalquantum.Thefeedbackschemeproposedheremeasurestheparticlepositionratherthanitsenergyandtriestoachievecoolingasphase-spacelocalisation,usingtheparticleoscillatorymotiontomixpositionandmomentumquadratures.

AsecondanalogyisgivenbythefactthattheproposedmethodneedsaDopplerpre-coolingstage,asithappensforresolvedsidebandcooling.Infact,theeffectivetrappedparticlepositionmeasurementisrealizedonlyintheLamb-Dickeregime,i.e.whentherecoilenergyismuchsmallerthantheenergyofavibrationalquantum,whichcanbeobtainedonlywhentheparticlehasundergoneapreliminarycoolingstage.Ourschemewillprovidethereforefurtherphasespacelocalisationandcooling.

Thepaperisorganizedasfollows.InsectionIIweshowhowtorealizetheindirectcontinuousmeasurementofthepositionbycouplingthetrappedparticlewithastandingwave.InsectionIIIweshallintroducethefeedbackloop,insectionIVweshallstudythepropertiesofthestationarystateinthepresenceoffeedbackandsectionVisforconcludingremarks.

II.CONTINUOUSPOSITIONMEASUREMENT

Weconsideragenericparticletrappedinaneffectiveharmonicpotential.Forsimplicityweshallconsidertheone-dimensionalcase,evenifthemethodcanbeinprinciplegeneralizedtothethree-dimensionalcase.Thisparticlecanbeaniontrappedbyalinearrf-trap[7]oraneutralatominanopticaltrap[3,8].Ourschemehoweverdoesnotdependonthespecifictrappingmethodemployedandthereforeweshallalwaysreferfromnowontoagenerictrapped“atom”.

Thetrappedatomofmassm,oscillatingwithfrequencyνalongthexˆdirectionandwithpositionoperatorx=

†1/2

x0(a+a),x0=(¯h/2mν),iscoupledtoastandingwavewithfrequencyωb,wave-vectorkalongxˆandannihilationoperatorb.Thestandingwaveisquasi-resonantwiththetransitionbetweentwointernalatomiclevels|+󰀕and|−󰀕.TheHamiltonianofthesystemis[9]

H=

hω0¯

󰀎

,whereω∼ωbwillbespecifiedlater,andmakingtherotatingwaveapproximation,h∆¯

thisHamiltonianbecomes

2

H=

2

(2σ−Dσ+−σ+σ−D−Dσ+σ−),

(5)

whereκisthespontaneousemissionrate.Herewehaveneglectedtherecoilandtheassociatedheatingofthevibra-tionalmotion.ThisisreasonableintheLamb-Dickelimitwehaveassumedfromthebeginning,sincetheassociated

heatingrateisgivenbyκ(kx0)2vibrationalquantapersecond,whichisnegligibleforasufficientlysmallLamb-Dickeparameterkx0.Inpracticalsituations,alsootherheatingmechanismsexist,causedbytechnicalimperfectionssuch

2

asthefluctuationsoftrapparametersduetoambientfluctuatingelectricalfieldsintheiontrapcase[7],andduetolaserintensitynoiseandbeam-pointingfluctuationsinthecaseoffar-offresonanceopticaltraps(seeRef.[8]andreferencestherein).Weassumethepresenceofthisheatingduetotrapimperfections,andwedescribeitwiththefollowingterminthemasterequation,characterizedbyaheatingrateγh(see[12])

LhD=

γh

2

󰀎󰀁†

2aDa−aa†D−Daa†.

(6)

Theheatingrateγhhasnottobetoolarge,inordertostaywithintheassumedLamb-Dickeregime.TheLamb-Dickeconditionalsoimpliesthatthetrappedatomhastobeinitiallypreparedinasufficentlycoldstate,i.e.,aneffectivethermalstatewith,say,ameanvibrationalnumbern0∼10.ThiscanbeobtainedwithapreliminaryDopplercoolingstage,whichisthenturnedoffatt=0andreplacedbytheproposedfeedbackcoolingscheme.Weshallseethatourschemeisabletofurthercoolthetrappedatom,closetothegroundstate,eveninthepresenceofmoderateheatingprocesses.

Theresultingmasterequationfortheinternalandvibrationaldegreesoffreedomis

˙=LhD−iD

2

(2σ−Dσ+−σ+σ−D−Dσ+σ−).

(7)

Letusnowseehowtorealizethecontinuouspositionmeasurement.Ithasbeenrecentlyshownthatwhenexcited

byalowintensitylaserfield,asingletrappedatomemitsitsfluorescentlightmainlywithinaquasi-monochromaticelasticpeak[13].Thefluorescentlightspectrumwasmeasuredbyheterodynedetection.Byimprovingthetechniqueitdoesnotseemimpracticaltogetahomodynedetectionofthesingle-ionfluorescentlight.InRef.[14],itwasshownhowonecouldachievesuch󰀁ameasurement.󰀎Thus,byexploitingtheresonancefluorescenceitcouldbepossibletomeasurethequantityΣϕ=σ−e−iϕ+σ+eiϕthroughhomodynedetectionofthefieldscatteredbytheatomalongacertaindirection[9].Infact,thedetectedfieldmaybewrittenintermsofthedipolemomentoperatorforthetransition|−󰀕↔|+󰀕as[9]

√(+)

Es(t)=

ηκξ(t),(9)

wherethephaseϕisrelatedtothelocaloscillator,which,sincewehaveassumedtheresonancecondition∆=0,in

thepresentcaseisprovidedbythesamedrivingfieldgeneratingtheclassicalstandingwave.ThesubscriptcinEq.(9)denotesthefactthattheaverageisperformedonthestateconditionedontheresultsofthepreviousmeasurementsandξ(t)isaGaussianwhitenoise[15].Infact,thecontinuousmonitoringoftheelectronicmodeperformedthroughthehomodynemeasurement,modifiesthetimeevolutionofthewholesystem,andthestateconditionedontheresultofmeasurement,describedbyastochasticconditioneddensitymatrixDc,evolvesaccordingtothefollowingstochasticdifferentialequation(consideredintheItosense)

˙c=LhDc−iD

√+

2

(2σ−Dcσ+−σ+σ−Dc−Dcσ+σ−)

κ

(Xρc⊗|+󰀕󰀔−|−ρc⊗|−󰀕󰀔+|X),(10)

whereρc=TrelDcisthereducedconditioneddensitymatrixforthevibrationalmotion.Intheadiabaticregime,theinternaldynamicsinstantaneouslyfollowsthevibrationaloneandthereforeonegetsinformationonthepositiondynamicsXbyobservingthequantityΣϕ.TherelationshipbetweentheconditionedmeanvaluesfollowsfromEq.(10)

3

󰀔Σϕ(t)󰀕c=

χ

+

󰀆

2κ

[X,[X,ρc]]

∆

sin(kx+φ)bψ−.(15)

Insertingthisequationinto(14),onegetsanequationforψ−whichisequivalenttohavethefollowingeffectiveHamiltonianforthevibrationalmotionoftheatomandthestandingwavemodealone

H=h¯δbb+h¯νaa−¯h

†

†

ǫ2

2∆

󰀊

bb+h¯νaa−¯h

††

ǫ2

∆

kx(|β|2+β∗b+βb†).

(18)

Shiftingtheoriginalongthexdirectionbythequantityh¯ǫ2k|β|2/∆mν2,onefinallygetsaneffectiveHamiltonian

analogoustothatoftheresonantcase(4)

H=h¯νa†a+h¯χYX,

4

(19)

wherenowχ=−4|β|kx0ǫ2/∆andtheatomicpolarizationσxisreplacedbythestandingwavefieldquadratureY=(be−iφβ+b†eiφβ)/2,whereφβisthephaseoftheclassicalamplitudeβ.Thismeansthatinthiscasethe“meter”isrepresentedbythecavitymode,andthataneffectivecontinuousmeasurementofthepositionofthetrappedatomisprovidedbythehomodynemeasurementofthelightoutgoingfromthecavity.ThismeasurementallowsinfacttoobtainthequantityYϕ=(ae−iϕ+a†eiϕ)/2,whichisanalogoustothequantityΣϕofthepreviousSection.Therefore,allthestepsleadingtoEq.(12)intheprecedingsubsection,canberepeatedhere,withtheappropriatechanges.Inthisnon-resonantcase,Dnowreferstothedensitymatrixofthesystemcomposedbyvibrationalmodeandthestandingwavemodeandthespontaneousemissionterminthemasterequation(7)hastobereplacedbytheformallyanalogoustermdescribingdampingofthestandingwavemodeduetophotonleakage.Thisisequivalenttointerprettheparameterκasacavitymodedecayrateinthiscaseandtoreplaceσ−withb,σ+withb†,andΣϕwithYϕinEqs.(7),(8),(9)and(10).Itisagainreasonabletoassumethatthestandingwavemodeishighlydamped,i.e.κ≫χ,sothatitispossibletoeliminateitadiabatically.Theperturbativeexpansion(10)nowbecomes

󰀈

χ2

(Xρc⊗|1󰀕󰀔0|−ρcX⊗|0󰀕󰀔1|)Dc=ρc−

κ

χ2

√+

κ2

ηχ

Kρc(t),

(21)

whereτisthetimedelayinthefeedbackloopandKisaLiouvillesuperoperatordescribingthewayinwhichthefeedbacksignalactsonthesystemofinterest.

Thefeedbackterm(21)hastobeconsideredintheStratonovichsense,sinceEq.(21)isintroducedaslimitofarealprocess,thenitshouldbetransformedintheItosenseandaddedtotheevolutionequation(12).Asuccessiveaverageoverthewhitenoiseξ(t)yieldsthemasterequationforthevibrationaldensitymatrixρinthepresenceoffeedback.Onlyinthelimitingcaseofafeedbackdelaytimemuchshorterthanthecharacteristictimeoftheamode,itispossibletoobtainaMarkovianequation[17,18],whichisgivenby

ρ˙=Lhρ−iνaa,ρ−

󰀋

†

ThesecondtermoftherighthandsideofEq.(22)istheusualdouble-commutatortermassociatedtothemeasurement

ofX;thethirdtermisthefeedbacktermitselfandthefourthtermisadiffusion-liketerm,whichisanunavoidableconsequenceofthenoiseintroducedbythefeedbackitself.

Then,sincetheLiouvillesuperoperatorKcanonlybeofHamiltonianform[17],wechooseitasKρ=−ig[P,ρ]/2[16],whereP=(a−a†)/2iistheadimensionalmomentumoperatorofthetrappedparticleandgisthefeedbackgainrelatedtothepracticalwayofrealizingtheloop.Onecouldhavechosentofeedthesystemwithagenericphase-dependentquadrature;however,itispossibletoseethattheabovechoicegivesthebestandsimplestresult[16].UsingtheaboveexpressionsinEq.(22)andrearrangingthetermsinanappropriateway,wefinallygetthefollowingmasterequation:

󰀂

χ2

2ηχ2/κ

ρ.(22)

5

ρ˙=−−

ΓΓg

2

2

󰀎󰀁

M∗2aρa−a2ρ−ρa2

󰀎󰀁

N2a†ρa−aa†ρ−ρaa†

gsinϕ

gsinϕ

󰀌

χ2

󰀌

γh+

χ2

4ηχ2/κ󰀍

i−

4ηχ2/κ

󰀍

−

1

+iνλ∂λ+2󰀅󰀈

Γ

=−ΓN|λ|2+

󰀊󰀈

sinϕ(λ∂λ∗+λ∗∂λ)C(λ,λ∗,t)2

󰀊󰀈󰀊󰀇

Γ

sinϕ(λ∗)2+sinϕλ2C(λ,λ∗,t),44

Γ

󰀇

(27)

Thestationarystateisreachedonlyiftheparameterssatisfythestabilityconditionthatalltheeigenvalueshave

positiverealpart,whichinthepresentcaseisachievedwhengsinϕ<0.Inthiscasethestationarysolutionhasthefollowingform

󰀍󰀌

1

(28)µ∗λ2,C(λ,λ∗,∞)=exp−ζ|λ|2+

2where

ζ=

N(g2sin2ϕ+4ν2)+gsinϕ(2νIm{M}−gsinϕRe{M})+g2sin2ϕ/2

+i

gsinϕ

4ν2

thatis,thestationarystateisaneffectivethermalstatewithmeanvibrationalnumberN.Thiscanalsobeseenfromthefactthatinthelargeνlimit,onecanconsiderthemasterequation(23)intheframerotatingatthefrequencyν,andneglectingtherapidlyoscillatingterms,oneendsupwithathermalmasterequationgivenbythefirstlineofEq.(23),whosesteadystateisjustthethermalstatewithmeanphononnumberN.

TheexpressionforNgivenbyEq.(24)showsthatitisconvenienttochooseϕ=−π/2togetthesmallestpossiblevaluesforN.Inthiswaythestabilityconditionisalsoautomaticallysatisfied.Then,theminimumvalueforthestationarymeanvibrationalnumberNcanbeobtainedbyminimizingitwithrespecttothefeedbackgaing:the

󰀁󰀁󰀎󰀎1/2

optimalvalueforgisgivenbyg=4γh+χ2/4κηχ2/4κandthecorrespondingminimumvalueofNis

Nmin=

1

1+4κγh/χ2

etimesitsmaximumheight.Weseethatthefeedbackproducesarelevantcontractionoftheuncertaintyregion,whichbecomesalmostindistinguishablefromtheregioncorrespondingtothemotionalgroundstate(innerdottedlineinFig.1).Theouterdashedlinecorrespondsinsteadtotheinitialthermalstatewithmeanvibrationalnumbern0=10,preparedbytheDopplerpre-coolingstage.

Intheresonantcaseinwhichtheeffectivepositionmeasurementisrealizedthroughthehomodynemeasurementofthefluorescence,themeasurementefficiencyismuchlowerandgroundstatecoolingbecomesverydifficulttoachieve.However,thispositionmeasurementschemebasedonfluorescencebecomesnecessarywhenonecannotextractthelightoutofacavity,suchasinRef.[13],andonehastousethefluorescentlight.

V.CONCLUSIONS

Wehaveproposedafeedbackschemeabletoachievesignificantcoolingofthemotionaldegreeoffreedomofatrappedparticle.Themethodisbasedonaneffectivecontinuousmeasurementoftheparticlepositionwhichcanberealizedintwodifferentways:byhomodyningeitherthefluorescenceofastrongtransitionordirectlythelightexitingthecavitywhichiscoupledtothetrappedparticle.Whentheefficiencyofthehomodynedetectionisclosetoone,themethodisabletoachievegroundstatecooling.Itisinterestingtonotethat,evenifonlytheparticlepositionismeasuredandthefeedbackischoseninordertodecreasepositionfluctuations,theschemeprovidesaphase-spacelocalisationforallquadratures.ThisisessentiallyduetothefactthatthebareatomHamiltonianh¯νa†amixesthedynamicsoftheatomicpositionandmomentum,sothatthecontinuoushomodynemeasurementactuallygivesinformationsonbothquadratures.Thismodelsharessomepeculiaritieswiththatonewehaveproposedin[20]tocoolthevibrationalmotionofamacroscopicmirrorofanopticalcavity.Thepresentapplicationtoatrappedatomstressestheversatilityofthemethodsusingfeedbackloopsinsystemscharacterizedbytheradiationpressureforceincontrollingthermalnoise.

Itisalsopossibletoseethattheproposedschemeisnotabletoreducethenoisebelowthequantumlimit,i.e.thestationarystatevarianceofagenericmotionalquadraturecannotbemadesmallerthan1/4.ThisisagainaconsequenceofthefreevibrationalHamiltonianh¯νa†a(itisinfactpossibletogetpositionsqueezinginthelimitingcaseν=0[21]).Actually,positionsqueezingcanbeobtainedbyconsideringasuitablemodificationofthepresentscheme[22].

AsconcernsthespecificwayinwhichaparticularfeedbackHamiltoniancouldbeimplemented,theimportantpointistobeabletorealizeaterminthefeedbackHamiltonianproportionaltomomentum.Thisisnotstraightforward,butcouldberealizedbyusingthefeedbackcurrenttovaryanexternalpotentialappliedtotheatomwithoutalteringthetrappingpotential.Ontheotherhand,shiftsintheposition(beingstrictlyequivalenttoalinearmomentumtermintheHamiltonian),areachievedsimplybyshiftingallthepositiondependenttermsintheHamiltonian,inparticularthetrappingpotential.Alternatively,theuseoflaserpulsescouldbeusefulaswell,since,usingatypicallasercoolingscheme,thelightcanexertontheatomaforceproportionaltoitsmomentum.

7

Aswehavealreadyremarked,inprinciplethemodelcouldbeextendedtothethreedimensionalcase.AsconcernsthemodeldiscussedinSec.IIA,oneshouldconsiderthreedifferentinternaltransitions,eachonecoupledwithavibrationaldegreeoffreedom,resonantwiththreeorthogonalstandingwaves.Fortheoff-resonantcasepresentedinSec.IIB,oneshouldonlyconsiderthreeorthogonalstandingwavesfarfromresonanttransitions.

Inconclusion,althoughtheimplementationofthepresentedcoolingmethodviafeedbackcouldbeanhardtask,itcanbeusefulwhenevertheuseofresolvedsidebandcoolingisimpractical.Thepossibilityofhavinganalternativewaytocooltrappedparticlesisparticularlyinterestingforquantuminformationprocessingapplications,becauseitmayhappenthattherequirementofhavingtwohighlystableinternalstatesforquantumlogicoperationsandagoodinternaltransitionforsidebandcoolingcannotbesimultaneouslysatisfied.Withthisrespectothercoolingstrategieshavebeenrecentlyproposed,asforexampletheuseofsympatheticcoolingbetweentwodifferentspeciesofions[5,23].

ACKNOWLEDGEMENTS

ThisworkhasbeenpartiallysupportedbyINFM(throughtheAdvancedResearchProject“CAT”),bytheEuropeanUnionintheframeworkoftheTMRNetwork“MicrolasersandCavityQED”.

P21

X

-2-1

-1-2

12

√

FIG.1.PhasespaceuncertaintycontoursobtainedbycuttingtheWignerfunctionofthestationarystateat1/