Continuous stochastic Schrodinger equations and localization

arXiv:quant-ph/9708011v1 6 Aug 1997M.Rigo,F.Mota-FurtadoandP.F.O’Mahony

DepartmentofMathematics,RoyalHolloway,UniversityofLondon,Egham,SurreyTW200EX,UK

Abstract.Thesetofcontinuousnorm-preservingstochasticSchr¨odingerequationsassociatedwiththeLindbladmasterequationisintroduced.Thissetisusedtodescribethelocalizationpropertiesofthestatevectortowardeigenstatesoftheenvironmentoperator.Particularfocusisplacedondeterminingthestochasticequationwhichexhibitsthehighestrateoflocalizationforwideopensystems.Anequationhavingsuchapropertyisproposedinthecaseofasinglenon-hermitianenvironmentoperator.Thisresultisrelevanttonumericalsimulationsofquantumtrajectorieswherelocalizationpropertiesareusedtoreducethenumberofbasisstatesneededtorepresentthesystemstate,andtherebyincreasethespeedofcalculation.

1.Introduction

Aquantumsysteminteractingwithitsenvironmentcanbedescribed,intheMarkovianapproximation,bytwocomplementaryapproaches.Inthefirstandmostcommonlyused[1,2,3],thesystemisrepresentedbyadensityoperatoranditsevolutionisdescribedbyamasterequation.InthesecondastatevectorrepresentsthesystemandastochasticSchr¨odingerequationdescribesthestateevolution[4,5,6]Thesetwotreatmentsareequivalentinthefollowingsense:foralltimesanensembleofstatevectorsgeneratedbyastochasticSchr¨odingerequationreproduces,onaverage,thedensityoperatorgeneratedbythemasterequation.Thecorrespondenceisnotuniquelydefined,inthatforasinglemasterequationtherearemanyassociatedstochasticequations.

Stochasticstatevectorequations,alsocalledunravelingsofthemasterequation,havebeenintroducedindifferentcontextsandwithdifferentinterpretations.Inthefundamentaltheoryofquantummeasurement,stochasticequationshavebeenusedtodescribethegeneraldynamicalprocessofthestatecollapseintoaneigenstateofthemeasuredobservable,i.e.localization[4,7,8,9,10,11,12,13].Inquantumoptics,stochasticSchr¨odingerequationshavebeenusedtodescribethesystemstateconditionedbymeasurementoutcomes.Inthiscontext,anunravelingcorrespondsto

2

aspecifiedmeasurementscheme,suchasphotoncounting,heterodyneorhomodynedetection[6,14].Moregenerally,inthefieldofopenquantumsystems,unravelingshavebeenusedasanefficientnumericalmethodtosolvethemasterequation[15,16,17,18,19,20,21,22,23,24,25,26].

Thepresentworkismotivatedbyarecentstudyofaquantumsystemininteractionwithathermalbathusingthequantumjump(QJ)unraveling[27].Itisshownthat,undersomeassumptionsvalidintheclassicallimit,theQJtrajectories,i.e.,therealizationofthestochasticprocess,approachadiffusivelimitverysimilartotheoneexhibitedbythequantumstatediffusion(QSD)trajectories.Sincediffusionisexpectedtobeageneralfeatureassociatedwiththeemergenceofclassicality,adescriptionofthewholesetofcontinuousunravelingsbecomesimportant.Thissetisintroducedinaunifiedwayinthefollowingsection.Wewillshowthateachcontinuousunravelingcanbecharacterizedverysimplybyspecifyingitsnoisecorrelations.Thesetofcontinuousunravelingsisthenusedtostudyhowquantumstateproperties,suchaslocalization,evolvewhentheunravelingchanges.

Averyimportantcharacteristicofquantumtrajectorieswhichhasbothphysicalandnumericalconsequencesisthelocalizationofquantumstatestowardeigenstatesoftheenvironmentoperator.Workingwiththerealnoiseunraveling(RN),Gisin[7]hasshownthatforanarbitraryhermitianLindbladoperatorLthestatevectorsconcentrateontheeigenspaceofL.Percival[9]extendedthisresultbygivingaproperdefinitionoftheensemblelocalizationofanarbitraryoperatorandthenprovidinganalyticalboundsfortherateofself-localizationofhermitianandnonhermitianLindbladoperatorsforthequantumstatediffusionunraveling.Foradissipativeinteraction,GarrawayandKnight[28,29]havepresentednumericalsimulationsofthelocalizationprocessusingtheQJandQSDunravelings.Startingfromdifferentquantumstates,suchasasuperpositionoftwocoherentstates,aFockstateandasqueezedgroundstate,theyhaveshownthatsuchstatesarehighlysensitivetodissipation.Theyalsoillustratedthelocalizationprocess.RecentlytheyhaveappliedtheirresultstodescribetheevolutionofaSchr¨odingercatstateinaKerrmediumwherelocalizationcompeteswithnonlineareffects[30](seealso[31]).

Fornumericalsimulationofopenquantumsystems,individualtrajectorieshaveprovenadvantageousoverdensityoperatorcomputations.Themainadvantagesstemfromthefactthatlessspaceisneededtostoreandpropagateintimeastatevectorthanadensitymatrix.Inaddition,fortrajectorymethodsonecanexploitthelocalizationpropertytoreducethenumberofbasisstatesneededtorepresentthestatevector,thussignificantlyreducingthetimeneededtocalculatequantumtrajectories.Forquantumjumpunravelings,whenmanyLindbladoperatorsarepresent,itiswellknownthatonecanperformaunitarytransformationtoselectoneofthequantumjumpsunravelings,insuchawayastominimizethenumberofbasisstatesneeded(seeref.[22]foran

3

applicationofthisproperty).ThelocalizationofthestatevectorforQSDhasbeenexploitedinthemixedrepresentationofSteimleandal.[23]andthemovingbasisofSchackandal.[24,25,26].

Insection3weusethesetofcontinuousunravelingstodescribelocalizationpropertiesforasingleenvironmentoperator.Severalwellknownresultsarerecoveredforahermitianoperator[4,32].Inthecaseofanon-hermitianoperatortheminimalrateoflocalizationintroducedfortheQSDunraveling[9,11,12]isextendedtothecompletesetshowingthatlocalizationisageneralfeaturesharedbyallthecontinuousunravelings,andanewunravelingisintroduced.Sometheoreticalargumentssupportedbynumericalsimulationssuggestthatthisnewunravelingpossessesthehighestlocalizationrate.

Inthepresentworkwemakeuseofthefreedomofchoiceforthenoisecorrelationstoobtainthecontinuousunravelingwhichlocalizesthestatevectorthemost.Thistransformationshouldnotbeconfusedwiththeunitarytransformationdiscussedabove.Thesetwotransformationsarecomplementaryandcanbeusedtogether.Attheendofsection3wecomparethelocalizationpropertiesofQSDandofourproposedunraveling.

Finally,insection4wesummarizeourresultsanddrawconclusionsabouttheapplicabilityofthesetofcontinuousunravelingstothestudyofthequasi-classicallimitofopenquantumsystems.2.

Continuousunravelings

WeproceedfollowingcloselythederivationofthequantumstatediffusionunravelingbyGisinandPercival[8].Inthiswork,ageneralstochasticdifferentialequationwithacomplexWienerprocessisusedasastartingpoint.ThedriftandnoisetermsarethenspecifiedbyaskingthatthestochasticdifferentialequationrecoversonaveragetheLindbladmasterequationforthedensityoperatorρofthesystem

ρ˙=−

i

2

{L†jLj,ρ}

󰀆

,(1)

whereHisthesystemHamiltonianandLj(j=1,...,J)thesetofLindbladoperators

whichrepresenttheinfluenceoftheenvironment.(Sincethemasterequation(1)isvalidunderaMarkovianapproximation,allthestochasticdifferentialequationsconsideredapplyonlywithinthisapproximation).Theotherconditionsneededtospecifythedriftandnoisetermsarethatthestateremainsnormalizedandthatthestochasticequationsharesthesameinvariancepropertiesunderunitarytransformationsasthemasterequation.ThislastconstraintisusedtoprovetheuniquenessofQSDamongthesetofcontinuousunravelings.RemovingtheconstraintofinvarianceunderunitarytransformationsamongtheLindbladoperators,weobtainthesetofcontinuousnorm-preservingunravelingsrelatedtothemasterequation.

4

2.1.DerivationofthestochasticSchr¨odingerequations

WestartbyconsideringageneralstochasticdifferentialequationofthefollowingItˆoformwhichgivesthevariation|dψ󰀑ofthestatevector|ψ󰀑inatimedt

|dψ󰀑=|v󰀑dt+

J󰀊

j=1

|uj󰀑

󰀇

n=1

N󰀊

αjndWjn

󰀈

(2)

where|v󰀑and|uj󰀑arevectors,αjnarecomplexnumbersanddWjnindependentreal

Wienerprocesses[33]whichobeythefollowingrelationships

M(dWjn)=0

dWjndWkm=δjkδmndt

dWjndt=0

(3)

whereMrepresentstheensembleaverage.Thetwoconditionstoberespectedbythepreviousequation(2)are(i)thestateisnormalizedforalltimes󰀒ψ|ψ󰀑t=1and(ii)foreachtime,themeanoftheprojectorassociatedtothestate|ψ󰀑givesthedensitymatrixρ=M(|ψ󰀑󰀒ψ|)withthedensitymatrixρevolvingaccordingtothemasterequationinLindbladform(1).Inthefollowing,thesetwoconditionswillbeusedtorelatethedriftterm|v󰀑andthestochasticterm|uj󰀑tothestate|ψ󰀑aswellasgivingconstraintsonthecomplexnumbersαjn.Noticethattheαjnmayalsodependonthestate|ψ󰀑andontimet.

ByfollowingcloselytheQSDderivationgiveninreference[8],weobtaintheexpressionforthedriftterm

|v󰀑=−

i

2

󰀊󰀁

j

L†jLj

+

󰀒L†j󰀑ψ󰀒Lj󰀑ψ

−

2󰀒L†j󰀑ψLj

󰀄

|ψ󰀑(4)

whichdiffersfromtheQSDderivationbytheintroductionofthenormalizationfactor

󰀃

(n|αjn|2)−1/2andthesetofcomplexnumbersβjk.ThelatterarearbitrarycoefficientsofaJ×Junitarymatrixwhicharisesduetothefreedomofchoiceinthelinearcombinationofvectors(Lk−󰀒Lk󰀑ψ)|ψ󰀑usedtoexpress|uj󰀑.

Finallyonegetstheequationforthestatevectorincrement

|dψ󰀑=−

+i

J󰀊

where󰀒Lj󰀑ψ=󰀒ψ|Lj|ψ󰀑istheexpectationvalueofLjforthestate|ψ󰀑.Thedriftterm

|v󰀑isthesameasthatobtainedintheQSDderivation,butthestochasticvectors|uj󰀑become

󰀊1

|uj󰀑=󰀃βjk(Lk−󰀒Lk󰀑ψ)|ψ󰀑j=1...J(5)

2

kn|αjn|

2j=1

k=1

J󰀁󰀊

L†jLj

+

󰀒L†j󰀑ψ󰀒Lj󰀑ψ

−

2󰀒L†j󰀑ψLj

󰀄

|ψ󰀑dt

(6)

(Lk−󰀒Lk󰀑ψ)|ψ󰀑dζk

5

Thisequationshowsthatalltheindeterminacyduetothecoefficientsαjnandtotheunitarytransformation(βjk)canbeincludedinthenoisetermsdζjwhicharegivenby

dζk=

J󰀊

βjk

j=1

󰀃

n

ItcanbeseeneasilythattheyhavezeromeanM(dζj)=0andcorrelations

∗

dζjdζk=δjkdt

󰀃

αjndWjn

n

|αjn|2

(7)

anddζjdζk=cjkdt(8)

wherecjkarecorrelationcoefficientsrelatedtotheunitarytransformation(βjk)andthe

noisecoefficientsαininthefollowingway

cjk=

J󰀊i=1

βijβikciwithci=

󰀃

n

2

αin

2forallj.Inthisspecialcasethecorrelationscjkvanish.

2.2.Unitarytransformation

LetusintroducethefollowingunitarytransformationamongLindbladoperators

Lj=

󰀊

k

˜k−λj1ujkL1

(10)

6

whereujkandλjarecomplexnumbersand(ujk)aunitarymatrix[8,9,35].Withthis

˜k=󰀃jujkdζjwiththecorrelationstransformationthenoisetermsbecomedζ

˜jdζ˜∗dζk

=δjkdt

and

˜jdζ˜k=dζ

m,n=1J󰀊

umjunkcmndt(11)

Thesecorrelationswilldependontheunitarytransformation(ujk)unlessallthe

correlationfactorsvanish,i.e.cjk=0.Since(βjk)isitselfaunitarytransformation,anecessaryconditionforinvarianceunderunitarytransformationisgivenby

cj=0forallj=1,...,J.

(12)

WhenonlyoneWienerprocessN=1ispresent,theunitaryinvariancecondition(12)impliesαj1=0forallj.AsaconsequencethereisnoinvariantunravelingwithonlyoneWienerprocess.WithtwoWienerprocessesN=2,theinvarianceconditionbecomes22αj1+αj2=0.Thenormofthetwocomplexnumbersisthesame|αj1|=|αj2|andthephasesarerelatedbyφj1−φj2=π/2+nπwherenisanyintegernumber.Thesimplestcasen=0leadsto

dζj=eiφj

󰀇

dWj1+idWj2

2

󰀈

(13)

whichcorrespondtothecomplexnoiseusedintheQSDunravelingwhenthephasesφjaresettozero.Thesimplestcasewhichcansatisfytheinvariancecondition(12)isgivenbytheQSDunraveling.Thephasesφjandotherchoicesofnintroduceonlyirrelevantphasefactorswhichcanbeneglected.ThisistheuniquenessresultofGisinandPercivalforQSD.IfoneconsidersmorethantwoWienerprocessesN≥3,itispossibletoconstructotherunravelingsinvariantunderunitarytransformation.Forinstance:

dW1+eiπ/3dW2+e−iπ/3dW3

dζ=

3

(14)

wherewehaveomittedtheindexjandthephasefactor.Sincealltheseunravelingshavethesamecorrelationsdζ2=0and|dζ|2=dt,theyareequivalent[33]andcanbereplacedbytheQSDunraveling.3.

Localization

Asanapplicationofthesetofcontinuousunravelingobtainedinthepresentwork,onecancomputetherateofself-localizationofasingleenvironmentoperatorLforawideopensystem,i.e.H=0,anddeterminetheeffectofthenoisecorrelationontherateofself-localization.Therateofself-localizationisdefinedastherateatwhichtheensembleaverageofthequantummeansquaredeviationdecays[9].Itisalsothe

7

ensembleaveragerateatwhichthestatevector|ψ󰀑tendstowardsoneofthe(right-)eigenstatesoftheLindbladoperator.Thequantummeansquaredeviation†oftheoperatorLisdefinedasσ2(L)=󰀒L†L󰀑ψ−󰀒L†󰀑ψ󰀒L󰀑ψ.Moregenerallythequantumcovarianceoftwooperatorsforthestate|ψ󰀑isσ(A,B)=󰀒A†B󰀑ψ−󰀒A†󰀑ψ󰀒B󰀑ψ[9].NotethatthequantumcovarianceofLwithitselfisjustthequantummeansquaredeviationσ2(L)=σ(L,L).Werestrictourattentiontoawideopensystembecausewewanttodescribethelocalizationprocess,independentlyoftheactionoftheHamiltonian.This,clearly,isonlyafirststeptowardsaproperunderstandingoflocalizationwhichshouldinvolveHamiltonianeffectsaswell.3.1.Hermitianenvironmentoperator

ForawideopensystemwithahermitianenvironmentoperatorL=L†thestatevector|ψ󰀑evolvesaccordingto

|dψ󰀑=−

1

(18)=−2(1+Re(c))Mσ(L)

dt

Thenoisecorrelationcisacharacteristicsignatureofthechosenunraveling.Forc=0,thequantumstatediffusionresult,givingaminimallocalizationrateof2,isrecovered[9].Inthiscase,asinalmostallcases,themeansquaredeviationtendstozero,thusthestate|ψ󰀑evolvestowardsoneeigenstateofL.Therealnoise(RN)unravelingc=1isclearlytheonewhichgivesthehighestrateofself-localization.Asaconsequence,fornumericalsimulationsinvolvinganarbitraryHamiltonianandonehermitianenvironmentoperator,theRNunravelingshouldbeusedsinceitwillproduce

†Notethatthequantummeansquaredeviationisnotanensembleaverage.

󰀁

22

󰀄

8

thefastestlocalization(forcontinuousunravelings).IntheoppositecasetotheRNunraveling,ifoneusestheimaginarynoiseunravelingc=−1,themeanlocalizationratevanishesandthestatedoesnotevolvetowardsaneigenstateofL.Recoveringthesewellknownresults[4,32]confirmsthevalidityofequation(15).3.2.Nonhermitianenvironmentoperator

WeconsiderthecaseofasinglenonhermitianLindbladoperator.Sincethiscaseismoredifficulttotreat,werestrictourselvestothemorespecificcaseofanannihilation

√

operatorL=

κRe

†

whichinvolvesthequantumcovarianceσ(a†,a)=󰀒a2󰀑ψ−󰀒a󰀑2ψ.Theequationforσ(a,a)canalsobederivedtogive

󰀉󰀁

σ(aa,a)−󰀒a󰀑ψσ(a,a)−󰀒a󰀑ψσ(a)dζ

†††

2

󰀄󰀋

(19)

dσ(a,a)=−κ(1+2σ(a))σ(a,a)+cσ(a,a)+cσ(a)

√+2

†

󰀁

2

††

2

∗2

2

󰀄

dt

thethirdtermofthisexpressionbeingpositivesince|c|≤1.Asaconsequence,theargumentforglobalstabilityofcoherentstates,

Mdσ2(a)

|σ(a†,a)|

}(21)

󰀈

9

independentofthechoiceofunraveling.Thisresultshowsthat,foranannihilationoperator,allunravelingslocalizeandκprovidesaminimalbound,independentoftheunraveling,fortheensemblemeanlocalizationrate.

Forahermitianoperator,theunravelingwhichlocalizesthemostwaseasytofindsincetheevolutionofthequantummeansquaredeviationisnotcoupledtoanyothermoment.Furthermorethecorrelationfactorcfactorises,makingtheunravelingindependentofthestate.Inthepresentcasethesituationismorecomplex,sincenoneofthesetwosimplifyingconditionsaresatisfied.Inthecaseofanannihilationoperator,weadoptthefollowingtechnique.Insteadofconsideringthelocalizationofanarbitrarystate|ψ󰀑,werestrictourattentiontosqueezedstates.Wewillshowthatunraveling(15)withL=

√

every

1−|γ=γσtwheretheindexsspecifiesthat|2

ts(a†,a)∗

(24)

themeansquaredeviationistakenwithrespecttothesqueezedstate|γt,αtdependsonlyonthesqueezing󰀑.Thisparameter.

lastrelationtellsusthatthemeansquaredeviationAconditiontocheckifsqueezedstatesarepreservedcanbeobtainedbydifferentiating(23)[11,35].Inordertosimplifythecalculation,theStratonovichformalismisused.Inthisformalism,theusualdifferentiationrulesapply.Thusfrom(23),astate|ψ󰀑initiallysqueezedwillremainsqueezedifitispossibletowrite

(a−γta†−αt)|dψ󰀑=(dγta†+dαt)|ψ󰀑

(25)

where|dψ󰀑istobeexpressedinStratonovichform.Fromequation(15)andusingtheusualconversionformulaX◦dY=XdY+1

†2LL−2󰀒L†

󰀑ψ(L−󰀒L󰀑ψ)−

c

󰀉

󰀋

|ψ󰀑dt

10

+(L−󰀒L󰀑ψ)|ψ󰀑◦dζ

(26)

Insertingthisexpressioninthecondition(25),onefindsnotonlythatsqueezedstatesarepreservedbutalsotheequationsforthesqueezingparametersare

dγt=−κγt(1+cγt)dt

κ

dαt=−

(27)

κγtdζ

(28)

writteninItˆoform.Sincetheevolutionofthesqueezingparameterγtisdeterministic,itiseasytofindtheunravelingwhichproducesthefastestsqueezingdecay.Itisgivenbythefollowingcorrelationfactor

∗γt

c=

11

Ifthesystemstateisnotasqueezedstate,thepreviousderivationdoesnotapplyanymore.Nevertheless,wecantrytogeneralizetheresultforanarbitrarystate.Usingtherelation(24)betweensqueezingparameterandmeanquantumdeviation,thesameunravelingcanbespecifiedas

c=

σ(a†,a)∗

superpositionofcoherentstates(|α󰀑+|−α󰀑)/

√

2anda

12

2520M σ2(a)15105000.010.020.030.04κt0.050.060.07Figure1.Ensembleaverageofthequantummeansquaredeviationσ2(a)showingtheshorttimescalelocalization.TheinitialstateistheFockstate|24󰀑.Eachcurverepresentadifferentunraveling:theunraveling(31)(——),QSD(----)andRealNoise(—·—).Theensembleaverageiscomputedusing1000trajectories.Theerrorsbarsindicatethe95%-confidenceintervals.

2520M σ2(a)15105000.010.020.030.04κt0.050.060.07Figure2.Asfigure1,butwiththeinitialstateinasuperpositionoftwoFockstates2−1/2(|23󰀑+|25󰀑).

13

2015M σ2(a)105000.010.020.030.04κt0.050.060.07Figure3.Asfigure1,butwiththeinitialstateinasuperpositionoftwocoherentstates2−1/2(|α󰀑+|−α󰀑)withα=4.

10010M σ2(a)10.10.010.00100.511.5κt22.53Figure4.Asfigure2,butforalongertimescale.

14

acorrelationfactorgivenby

†c=

σ(L,L)∗

2(q)

|σ(q†,q)|

=

σ¯χa†2a2

subjecttodissipationL=

√

2

h(34)

15

1.2

(a)

10.80.60.40.20

0.2

Var{σ(a)}M σ2(a)0.6

21

(b)

0.8

0.4

01234

5λ

6710

0

01234

5λ

6710

Figure5.Ensembleaverage(a)andvariance(b)ofthequantummeansquaredeviationσ2(a)versusthescalingparameterλforthekickedanharmonicoscillator.TheuppercurvecorrespondstotheQSDunravelingandthelowercurvetothetimedependentunraveling(31).Theerrorsbarstakeintoaccountstatisticalaswellasnumericalaccuracyuncertainties.

Infigure5,theensembleaverageandvarianceofthequantumsquaredeviationσ2(a)arerepresentedfordifferentvaluesofthescalingparameterλ.Thevaluesrepresentedarestationaryresultsobtainedbyintegratingtheequationsofmotionsovertypically2500periodsandtakingthemeanoverasingletrajectory.Suchalongintegrationintimeisnecessaryinordertoobtainaproperaverageoverthestrangeattractorofthechaoticsystem.Thesystemparametersaresettothefollowingvaluesχ=1,β0=2,κ=0.5,τ1=0.98andτ2=1forwhichtheclassicalsystemisknowntobechaotic.Theprecisionofthenumericalresultsisestimatedbyrepeatingthecalculationsfordifferentnumericalparameterssuchasthetimestepsize.

Figure5(a)showsforbothunravelingsaslowlydecayingensembleaverageMσ2(a)foranincreasingvalueofthescalingλ.Noticethattheamplitudeofmotionrescalesasλthustheratioσ2(a)/M󰀒a†a󰀑tendstowardszerowhenλ→∞,providinganumericaljustificationfortheemergenceoftheclassicalattractor.Furthermore,fig5(a)showsthattheunraveling(31)reduces,comparedtotheQSDunraveling,thestationaryvalueofthemeansizeofthewavepacket.Thereductioncanbeupto20%dependingonthescaleparameterλ,thelargestreductionbeingachievedinthequantumregime.

Moreimportantisthereductionofthesizeofthefluctuationsshowninfig5(b).ThepicturesuggestedisthateachtimethewavepacketdeviatesfromacoherentstatetheQSDunravelingtendstorestoretheshapebyapplyingahomogeneousnoise,whiletheunraveling(31)adaptsbyapplyinganon-homogeneousnoiseinthedirectionofthelargestdeviation.ThisadaptabilitydoesnotproduceanimportantreductionofthewavepacketsizebutcanstabilizethewavepacketmoreefficientlyascomparedtoQSD.

1.

Discussion

WehaveintroducedthesetofcontinuousunravelingswhichrecoversinmeanthemasterequationinLindbladformandpreservesthenormofthestatevector.Thequantumstatediffusionunravelingisamemberofthisset,beingthesimplestwhichpreservesthesameinvariancepropertiesunderunitarytransformationsasthedensitymatrix.Wehaveseenthateachsingleunravelingcanbespecifiedverysimplybythechoiceofthenoisecorrelationsthusprovidinganaturalclassification.Fortheoreticalpurposes,itisusefultoworkwiththefullsetofcontinuousunravelingssinceitallowsonetostudyhowquantitieswhichdependonthechoiceoftheunravelingaresensitivetothischoice.

Asafirstapplication,wehavestudiedthelocalizationpropertieswhenonlyasingleLindbladoperatorispresent.Inthecaseofahermitianoperator,thehighestlocalizationrateoftherealnoiseunravelingaswellastheabsenceoflocalizationoftheimaginarynoiseunravelinghavebeenrecoveredandexplainedinaconsistentway.Foranonhermitianoperator,namelytheannihilationoperator,anewtimedependentunravelinghasbeenintroduced.Itisshownanalyticallythatthisunravelingprovidesthehighestlocalizationrateforsqueezedstatesandnumericallythatthispropertyisalsovalidformorecomplexquantumstates.Thisunravelingmaximizesthelocalizationbycontinuouslyadjustingthephasenoiseaccordingtotheshapeofthewavepacket.Thisstudyprovidesabetterunderstandingofthelocalization.Forinstance,theQSDunravelingisknowntohavegoodlocalizationpropertiesduetoitsinvariancecorresponding,insomesense,toahomogeneousdistributionofnoise.Wehaveseenthatthelocalizationratecanbeincreasedbymaximizingthenormofthenoisecorrelationfactorandadjustingcontinuouslyitsphase,thislastconstraintleadingtotheintroductionofatimedependentunraveling.

Sincethenewunravelingincreaseslocalizationitisagoodcandidatefornumericalsimulationsofquantumtrajectoriesandforthesolutionofthemasterequation.AnumericalcomparisonofthewavepacketsizeandfluctuationsbetweenQSDandthenewunravelingshowsthatthenewunravelingperformsbetterthanQSDbystabilizingthesizeofthewavepacket.

Inconnectionwiththestudyofthequantum-classicaltransition,arecentworkbyBrunetal[27]hasshownthattheQuantumJumpunravelingtendstoacontinuousunraveling.Itcanbeeasilyseenthatthisunravelingisamemberofthesetintroducedinthepresentpaper.Wehaveshownthatforasimplequantumsystemsubjecttodissipationallmembersofthesetofcontinuousunravelingslocalizewithaminimalrategivenbythedissipationrate,makinglocalizationageneralpropertyvalidforallunravelingsinsteadofonlysomeparticularones.

17

Acknowledgments

WethankGernotAlber,NicolasGisin,IanPercival,R¨udigerSchackandWalterStrunzforstimulatingdiscussions.WeacknowledgefinancialsupportfromtheEUunderitsHumanCapitalandMobilityProgramme.AppendixA.

Propertiesofthenoisecorrelations

InthecaseofalinearcombinationoftwoWienerprocessN=2,thenoisetermdζisspecifiedbythetwocomplexnumbersα1andα2whichwewriteasα1=ρ1eiφ1andα2=ρ2eiφ2.Thenoisecorrelationfactorbecomes

c=

󰀃

n2αn

2

ρ21+ρ2

UsingR=ρ2/ρ1andθ=2(φ2−φ1),thiscomplexnumbercanberewrittenas

c=e

2iφ11

+R2eiθ

18

[13]StrunzWTandPercivalICThesemiclassicallimitofquantumstatediffusion-aphasespace

approach,submittedtoJPhysA

[14]WisemanHMandMilburnGJ1993Phys.Rev.A471652

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