Quantum Stochastic Differential Equation

arXiv:math-ph/0512078v122Dec2005QuantumStochasticDifferentialEquationforUnstableSystemsV.P.BelavkinMathematicsDepartment,UniversityofNottinghamNottinghamNG72RD,UnitedKingdomP.StaszewskiInstituteofMathematics,PedagogicalUniversityofBydgoszcz85-072Bydgoszcz,PolandReceived10December1999Publishedin:J.Math.Phys.41No11,2000,7220–7233.AbstractAsemi-classicalnon-Hamiltonianmodelofaspontaneouscollapseofunstablequantumsystemisgiven.Thetimeevolutionofthesystembe-comesnon-Hamiltonianatrandominstantsoftransitionofpurestatestoreducedones,η7→Cη,givenbyacontractionC.Thecountingtrajec-toriesareassumedtosatisfythePoissonlaw.Aunitarydilationoftheconcractivestochasticdynamicsisfound.Inparticular,inthelimitoffrequentdetectioncorrespondingtothelargenumberlimitweobtaintheItˆo-Schr¨odingerstochasticunitaryevolutionforthepurestateofunstablequantumsystemprovidinganewstochasticversionofthequantumZenoeffect.1IntroductionandsummaryThedecayprocessisbyitsnaturediscontinuousandtakesplaceatrandominstantsoftime.Nevertheless,someauthorssucceededindescribingquantumunstablesystemsbyconsidering“smoothed”timeevolutionofunstablesystemsinthedynamicalsemigroupapproach.TheuseofoneparametercontractingsemigroupinaHilbertspace[1]–[4]forthedescriptionofthedynamicsofunstablequantumsystemSgeneralizesthelawofexponentialdecaysayingthatthenumberofparticlesinagivenstatewhichhavenotdecayeduptotisanexponentialfunctionoftime;n(t)=n(0)exp[−λt],λ0t≥0.LetHbeaHilbertspaceofS,letψ(0)∈Hdenotesaninitial(pure)stateofS.Itisassumedthatforanyt≥0thestateofSisgivenbyformulaψ(t)=V(t)ψ(0),(1.1)1wherethefamily{V(t),t≥0}ofboundedoperatorsonHsatisfiesthefol-lowingconditions:(a)kV(t)k≤1,t≥0,(b)V(0)=I,(c)V(t1+t2)=V(t1)V(t2),t1,t2≥0,(d)themapt7→V(t)isstronglycontinuous.Thestate(1.1)isnormalizedtotheprobabilityp(t)=kψ(t)koffindingthesystemundecayedatt,moreoverp(t)monotonicallydecreasesasthesemigroupiscontracting.ByvirtueofSz-Nagytheorem[5]thereisaunitarydilationofthedynamicsV(t)ontheHilbertspaceK=H⊕K,whereKdenotestheHilbertspaceoftheproductsofthedecay.LetusassumethatthedecayofthestateoftheunstablequantumsystemSisrepresentedbycompletelypositivemapI:T(H)→T(H)oftheform[6]Iρ(t)=Cρ(t)C∗,C∗C≤I,(1.2)whereIistheidentityoperatorinH,theHilbertspaceofS.Thenthetimeevolutionofthemixedstateofthesysteminquestionisgivenbystronglycontinuouscontractingsemigroupwiththegeneratoroftheform[7,8]dρλ(t)dt=−i[H,ρλ(t)]+λI−I
ρλ(t),(1.3)whereHdenotesthehamiltonianoftheunstablequantumsystemS,andλ0isthedecayratio.Themixedstateρλ(t)satisfyingthedynamicalevolutionequation(1.3)isnormalizedtothesurvivalprobabilityTrρλ(t)forwhichddtTrρλ(t)=Tr(C∗C−I)ρλ(t)≤0.(1.4)InSect.2wegiveasemi-classicalnon-Hamiltonianmodelofspontaneouscollapseofanunstablequantumsystem.TheHamiltoniantime-evolutionofthesystembecomesnon-Hamiltonianatrandominstantsoftransitionsη7→Cηofpurestatestoreducedones,givenbythecontractionC.Itisassumedthatthecountingtrajectories,consistedofinstantsofoccurrencesofthecollapse,aredistributedaccordingtothePoissonlaw.Wefindthetime-developmentoftheclassicalstatepropagatorVtinHintheformofItˆostochasticequationwithrespecttotheclassicalPoissonprocess.Consequently,weobtainnonmixingItˆostochasticequationsforpure(resp.mixed)statesoftheunstablequantumsystemS.Itisshownthattheaverageddensitymatrixcorrespondingtothestatisticalmixtureofcollapsedstatessatisfieseq.(1.3).Assumingthateachcollapseη7→CηslightlychangesthestateofS(I−C=λ−1RwithboundedRsatisfyingforlargeλtheconditionR∗R≤λ(R+R∗))wefindthecontractingsemigroupequationresultingfromthestochasticdynamicsinthelargenumberlimitλ→∞.InSect.3wegivethequantumstochasticrepresentationbVtoftheclassicalstochasticpropagatorVtinHasanoperator-valuedprocessintheHilbertspaceH⊗F,whereF=F+(L2(R+))istheBoseFockspaceoverthesingle-particlespaceofsquare-integrablecomplexfunctionsonR+.Tothisendweemploythegeneratingfunctionalmethoddescribedinthissection.2AsaunitarydilationofacausalcontractivecocycleVtinHcannotingen-eralbeobtainedfromacausalunitarystochasticcocycleUtinthesameHilbertspaceH,itisimpossibletofindaHamiltoniansemiclassicaldynamicsgivingthecontractivestochasticdynamicsoftheunstablequantumsystemasthere-ducedone.Therefore,weconsidertheunitarydilationofthecontractionCinanextendedHilbertspaceH⊗C2,thelattercanbeinterpretedastheHilbertspaceof“quantummeter”detectingthedeathorlifeoftheunstableparticle.TheunitarydilationofthecontractivestochasticcocycleVt,cf.[9],isthenrealizedasacausalunitarycocycleUtinaHilberttensorproductH⊗F•,whereF•=F+(C2⊗L2(R+)),theBoseFockspaceoveroneparticlespaceC2⊗L2(R+),Sect.4.Weconsidertwocasesoftheunitarydilation(4.1)SofCinH⊗C2:(a)withSintheformofHermitianblock-matrix(4.3-4),(b)non-Hermitianunitaryblockmatrix(4.22).Incase(a)wefindtheQSDEfortheunitaryevolutioninH⊗F•withrespecttothequantumstochasticPois-sonmatrixprocessofintensityλ.Incase(b)wefindthelimit(asλ→∞)oftheunitaryevolutionusingthegeneratingfunctionalmethoddescribedinSect.3.ThelimitingunitaryevolutioninH⊗F•hastheformofthediffusionQSDEwithrespecttothefieldmomentumprocessbeingquantumstochasticrepresen-tationofthestandardWienerp