Adiabatic Approximation in the Density Matrix Appr

arXiv:quant-ph/0201020v21Apr2002AdiabaticApproximationintheDensityMatrixApproach:Non-DegenerateSystemsA.C.AguiarPinto(1),K.M.FonsecaRomero(2),andM.T.Thomaz(1)(1)InstitutodeF´ısica-Univ.FederalFluminenseAv.Gal.MiltonTavaresdeSouzas/n.◦,CEP:24210-340,Niter´oi,R.J.,Brazil(2)UniversidadNacional,FacultaddeCiencias,DepartamentodeF´ısica,CiudadUniversitaria,Bogot´a,ColombiaAbstractWestudytheadiabaticlimitinthedensitymatrixapproachforaquantumsystemcoupledtoaweaklydissipativemedium.Theenergyspectrumofthequan-tummodelissupposedtobenon-degenerate.Intheabsenceofdissipation,thegeometricphasesforperiodicHamiltoniansobtainedpreviouslybyM.V.Berryarerecoveredinthepresentapproach.Wedeterminethenecessaryconditionsatisfiedbythecoefficientsofthelinearexpansionofthenon-unitarypartoftheLiouvillianinordertotheimaginaryphasesacquiredbytheelementsofthedensitymatrix,duetodissipativeeffects,begeometric.Theresultsderivedaremodel-independent.Weapplythemtospin12modelcoupledtoreservoiratthermodynamicequilibrium.11IntroductionSincethefascinatingworkbyBerryin’84[1]showingtheexistenceofgeometricphases(path-dependentphases)invectorstatesdrivenbyadiabaticperiodicHamiltonians,au-thorsintheliteraturehavelookedforgeometricphasesinotherphysicalcontexts.Inparticular,Joyeetal.[2]andBerry[3]independentlyshowedthatthetransitionproba-bilityofinstantaneouseigenstatesofnon-realHamiltoniansinthenon-adiabaticregimegetsanimaginaryphase.ThisimaginaryphasewasmeasuredbyZwanzigeretal.foratwo-levelsystem[4].Theappearanceofanimaginarycorrectiontothegeometricphaseinquantummodelscoupledtodissipativemediahasbeendiscussedintheliterature[5,6,7].Inthosereferences,thenon-unitaryevolutionofthequantumsystemisimplementedbyaphenomenologicalnon-hermitianHamiltonian.Thisphenomenologicalapproachhasbeenextensivelyappliedtothestudyofthepropertiesofopenquantumsystems[8,9].Recentlywehaveconsideredaspin12modelinthepresenceofanexternalmagneticfieldandcoupledtoaweaklydissipativemedium[10];thissystemprecesseswithconstantangularvelocityaroundafixedaxis.WeappliedtwoLindbladoperatorstorepresentthenon-unitarypartoftheLiouvilliansofthequantumsystemincontactwithtwodistinctreservoirs.Usingthemasterequationsforthesemodels,weconcludedthatthegeometricphasesacquiredbythespin12instantaneouseigenstatesoftheHamiltonianwerenotmodifiedbythepresenceofthedissipation.TheeffectiveresultoftheinteractionofthequantumsystemwiththereservoiristheshrinkingoftheBlochvector,whichcanbeusedtogiveageometricdescriptiontothedensitymatrix[11,12,13,14].Certainlyitisstillanopenquestionwhetherourresultsinreference[10]areofgen-eralnatureorparticulartothedissipativemodelsstudied.Weremindthattheyareinoppositiontotheonesderivedbythenon-hermitianHamiltonianapproach[5,6,7].Inordertoprovethattheresultsof[10]arevalidingeneralforreservoirsatequilibrium,wemusthaveamodel-independentapproach.Weshouldsaythatmasterequationsaretheproperwaytostudyanopenquantumsystem,whereasphenomenologicalnon-hermitianHamiltoniansaresupposedtogivea“bonefide”descriptionofanopensystemonlywhenthecoherenceofstatesisnotdestroyedbytheinteractionofthesystemwithitsneighbor-hood.Wepointoutthatthegeneralexpressionfortheimaginaryphaseacquiredbythenon-hermitianHamiltoniansisgivenbyeqs.(75)and(76)ofreference[8],fromwhichweconcludethatthisimaginaryphasehasbegeometricforanynon-hermitianHamiltonian,asderivedinreferences[5],[6]and[7]forsomespecificmodels.However,bytheendofreference[5]GarrisonandWrightaffirmthattheirresultshouldbecheckedoutbya2densitymatrixapproach.Inreference[10]thedensitymatrixalreadyappearsasourcentralobject.However,eventhereourexplanationisbasedonthetimeevolutionoftheinstantaneouseigenstatesofHamiltonian.Inthepresentpaper,werederivetheevolutionofaquantumsystemintheadiabaticapproximationdirectlyinthedensitymatrixformulation,whichisthenaturalapproachinthestudyofquantumsystemscoupledtodissipativeenvironment.TheAdiabaticTheoremdiscussedinreferences[15,16,17]appliestoquantumsystemsdrivenbyunitaryevolution.ItstatesthatifinitiallythesystemisinaneigenstateoftheHamiltonian,itstimeevolutionisfasten,ateachtime,toaninstantaneouseigenstateofHamiltonianwiththesameoriginalquantumnumbers.Therefore,itisanaturalchoicetowriteourdensitymatrixinthebasisoftheinstantaneouseigenstatesoftheHamiltonianandderiveitsadiabaticlimit.Thedynamicsoftheoperatordensityρ(t)isgivenbyaLiouville-vonNeumannequa-tion.IntheLiouvillianweaddanon-unitarytermtotakeintoaccounttheinteractionwithadissipativemedium,thatisdρ(t)dt=−i[H(t),ρ(t)]+LDρ(t),(1)whereH(t)isthetime-dependentHamiltonianofthequantumsystemandLDisasuper-operatorthatactsonρ(t)andisresponsibleforthenon-unitaryevolutionofthequantumsystem.InthebasisoftheinstantaneouseigenstatesoftheHamiltonianH(t)(i.e.,H(t)|ui;ti=Ei(t)|ui;ti),thedynamicalequationsoftheelementsofthedensitymatrixaredρHij(t)dt=Xl[dhui;t|dt]|ul;tiρHlj(t)+ρHil(t)hul;t|[d|uj;tidt]−−i(Ei(t)−Ej(t))ρHij(t)+hui;t|LDρ(t)|uj;ti,(2)whereρHij(t)≡hui;t|ρ(t)|uj;ti.NotallelementsofmatrixρHij(t)areindependent,duetotheconstraintsthatTr[ρ(t)]=1andρHij(t)beahermitianmatrix.Thetime-dependentHamiltonia