Alternative Derivation of the Hu-Paz-Zhang Master

arXiv:quant-ph/9508004v14Aug1995Imperial/TP/94-95/55ALTERNATIVEDERIVATIONOFTHEHU-PAZ-ZHANGMASTEREQUATIONOFQUANTUMBROWNIANMOTIONJ.J.Halliwell∗andT.Yu†TheoreticalPhysicsGroup,BlackettLaboratory,ImperialCollege,LondonSW72BZ,U.K.(June,1995)AbstractHu,PazandZhang[B.L.Hu,J.P.PazandY.Zhang,Phys.Rev.D45(1992)2843]havederivedanexactmasterequationforquantumBrownianmotioninageneralenvironmentviapathintegraltechniques.Theirmasterequationprovidesaveryusefultooltostudythedecoherenceofaquantumsystemduetotheinteractionwithitsenvironment.Inthispaper,wegiveanalternativeandelementaryderivationoftheHu-Paz-Zhangmasterequation,whichinvolvestracingtheevolutionequationfortheWignerfunction.Wealsodiscussthemasterequationinsomespecialcases.PACSNumbers:05.40.+j,42.50.Lc,03.65.BzTypesetusingREVTEX∗E-mail:[e-mail]†E-mail:[e-mail]ianmotion(QBM)modelsprovideaparadigmofopenquantumsystemsthathasbeenveryusefulinquantummeasurementtheory[1],quantumoptics[2]anddecoherence[3-5].OneoftheadvantagesoftheQBMmodelsisthattheyarereasonablysimple,yetsufficientlycomplextomanifestmanyimportantfeaturesofrealisticphysicalprocesses.CentraltothestudyofQBMisthemasterequationforthereduceddensityoperatoroftheBrownianparticle,derivedbytracingouttheenvironmentintheevolutionequationforthecombinedsystemplusenvironment.Avarietyofsuchderivationhavebeengiven[6-9].ThemostgeneralisthatofHu,PazandZhang[10,11],whousedpathintegraltechniquesandinparticular,theFeynman-Vernoninfluencefunctional.ThepurposeofthispaperistoprovideanalternativeandelementaryderivationoftheHu-Paz-ZhangmasterequationforQBM,bytracingtheevolutionequationfortheWignerfunctionofthewholesystem.II.MASTEREQUATIONFORQUANTUMBROWNIANMOTIONThesystemweconsideredisaharmonicoscillatorwithmassMandbarefrequencyΩ,ininteractionwithathermalbathconsistingofasetofharmonicoscillatorswithmassmnandnaturalfrequencyωn.TheHamiltonianofthesystemplusenvironmentisgivenbyH=p22M+12MΩ2q2+Xnp2n2mn+12mnω2nq2n!+qXnCnqn,(1)whereq,pandqn,pnarethecoordinatesandmomentaoftheBrownianparticleandoscil-lators,respectively,andCnarecouplingconstants.Thestateofthecombinedsystem(1)ismostcompletelydescribedbyadensitymatrixρ(q,qi;q′,q′i,t)whereqidenotes(q1,...qN),andρevolvesaccordingto˙ρ=−i¯h[H,ρ].(2)2ThestateoftheBrownianparticleisdescribedthereduceddensitymatrix,definedbytracingovertheenvironment,ρr(q,q′,t)=ZYn(dqndq′nδ(qn−q′n))ρ(q,qi;q′,q′i,t).(3)Theequationoftimeevolutionforthereduceddensitymatrixiscalledthemasterequation.Forageneralenvironment,Hu,Paz,andZhang[10]derivedthefollowingmasterequationbyusingpathintegraltechniques:i¯h∂ρr∂t=−¯h22M∂2ρr∂q2−∂2ρr∂q′2!+12MΩ2(q2−q′2)ρr+12MδΩ2(t)(q2−q′2)ρr−i¯hΓ(t)(q−q′)∂ρr∂q−∂ρr∂q′!−iMΓ(t)h(t)(q−q′)2ρr+¯hΓ(t)f(t)(q−q′)∂ρr∂q+∂ρr∂q′!.(4)Theexplicitformofthecoefficientsoftheaboveequationwillbegivenlateron.Thecoef-ficientδΩ2(t)isthefrequencyshiftterm,thecoefficientsΓ(t)isthe“quantumdissipative”term,andthecoefficientsΓ(t)h(t),Γ(t)f(t)are“quantumdiffusion”terms.Generally,thesecoefficientsaretimedependentandofquitecomplicatedbehaviour.WefinditconvenienttousetheWignerfunctionofthereduceddensitymatrix,˜W(q,p,t)=12πZdueiup/¯hρrq−u2,q+u2,t.(5)TakingtheWignertransformof(4),weobtain1∂˜W∂t=−1Mp∂˜W∂q+M[Ω2+δΩ2(t)]q∂˜W∂p+2Γ(t)∂(p˜W)∂p+¯hMΓ(t)h(t)∂2˜W∂p2+¯hΓ(t)f(t)∂2˜W∂q∂p.(6)1WebelieveEq.(2.48)inRef.[10]containssomeincorrectnumericalfactors.3Theinversetransformationof(5)isgivenbyρr(q,q′,t)=Zdpe−ip(q−q′)/¯h˜Wq+q′2,p,t!.(7)Ourstrategyforderivingthemasterequation(4)istoderivetheFokker-Plancktypeequation(6)fromtheWignerequationforthetotalsystem.ThemasterequationcanbeobtainedfromtheWignerequationforthesystembyusingthetransformation(7).Weshallmakethefollowingtwoassumptions:(1)Thesystemandtheenvironmentareinitiallyuncorrelated,ie.theinitialWignerfunctionfactorsW0(q,p;qi,pi)=Ws0(q,p)Wb0(qi,pi),(8)whereWs0andWb0aretheWignerfunctionsofthesystemandthebath,respectively,att=0.(2)TheheatbathisinitiallyinathermalequilibriumstateattemperatureT=(kBβ)−1.ThismeansthattheinitialWignerfunctionofbathisofGaussianform,Wb0=YnWbn0=YnNnexp{−2ωn¯htanh(12¯hωnβ)Hn},(9)whereHnistheHamiltonianofthen-thoscillatorinthebath,Hn=p2n2mn+12mnω2nq2n.(10)Inaddition,oneeasilyseethattheinitialmomentsofthebatharehqn(0)i=hpn(0)i=0,(11)hqn(0)qm(0)i=0(ifm6=n),(12)hpn(0)pm(0)i=0(ifm6=n),(13)hqn(0)pm(0)+pm(0)qn(0)i=0,(14)and4hq2n(0)i=¯h2mnωncoth(12¯hωnβ),hp2n(0)i=12¯hmnωncoth(12¯hωnβ).(15)FortheQBMproblemdescribedby(1)and(2),theWignerfunctionofthecombinedsystemplusenvironmentsatisfies∂W∂t=−pM∂W∂q+MΩ2q∂W∂p+Xn−pnmn∂W∂qn+mnω2nqn∂W∂pn!+XnCnqn∂W∂p+q∂W∂pn!.(16)Byintegratingoverthebathvariablesonthebothsidesoftheaboveequation,oneobtains∂˜W∂t=−pM∂˜W∂q+MΩ2q∂˜W∂p+XnCnZYidqidpiqn∂W∂p,(17)where˜W(q,p)isthereducedWignerfunctionanditfollowsfromEq.(3)that˜W(q,p)=Z+∞−∞YidqidpiW(q,p;qi,pi).(18)ThisdefinitionisequivalenttoEqs.(3)and(5).Thefirsttwotermsontheright-handsideoftheEq.(17)giverisetothestandardevolutionequationofthesystem.Thelasttermcontainsalltheinformationaboutthebehaviourofthesysteminthepresenceofinteractionwithenvironment.Inwhatfollows,weshalldemonstr