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Open quantum systems I: the Hamiltonian approach PDF

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Lecture Notes in Mathematics 1880 Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris · · S. Attal A. Joye C.-A. Pillet (Eds.) Open Quantum Systems I The Hamiltonian Approach ABC Editors Stéphane Attal Alain Joye InstitutCamille Jordan Institut Fourier Université Claude Bernard Lyon 1 Universitéde Grenoble 1 21 av. Claude Bernard BP 74 69622 Villeurbanne Cedex 38402 Saint-Martin d'Hères Cedex France France e-mail:[email protected] e-mail:[email protected] Claude-Alain Pillet CPT-CNRS, UMR 6207 Université du Sud Toulon-Var BP 20132 83957 La Garde Cedex France e-mail:[email protected] LibraryofCongressControlNumber:2006923432 MathematicsSubjectClassification(2000):37A60,37A30,47A05,47D06, 47L30,47L90, 60H10,60J25,81Q10,81S25,82C10, 82C70 ISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 ISBN-10 3-540-30991-8 SpringerBerlinHeidelbergNewYork ISBN-13 978-3-540-30991-8 SpringerBerlinHeidelbergNewYork DOI10.1007/b128449 Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:1)c Springer-VerlagBerlinHeidelberg2006 PrintedinTheNetherlands Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. Typesetting:bytheauthorsandSPI Publisher Services usingaSpringerLATEXpackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:11602606 V A 41/3100/ SPI 543210 Preface This is the first in a series of three volumes dedicated to the lecture notes of the SummerSchool”OpenQuantumSystems”whichtookplaceattheInstitutFourier in Grenoble from June 16th to July 4th 2003. The contributions presented in these volumes are revised and expanded versions of the notes provided to the students duringtheSchool. Closedvs.OpenSystems By definition, the time evolution of a closed physical system S is deterministic. It is usually described by a differential equation x˙ = X(x ) on the manifold M of t t possibleconfigurationsofthesystem.Iftheinitialconfigurationx ∈ M isknown 0 thenthesolutionofthecorrespondinginitialvalueproblemyieldstheconfiguration x atanyfuturetimet.Thisappliestoclassicalaswellastoquantumsystems.Inthe t classicalcaseM isthephasespaceofthesystemandx describesthepositionsand t velocities of the various components (or degrees of freedom) of S at time t. In the quantumcase,accordingtotheorthodoxinterpretationofquantummechanics,M is a Hilbert space and x a unit vector – the wave function – describing the quantum t state of the system at time t. In both cases the knowledge of the state x allows t to predict the result of any measurement made on S at time t. Of course, what we mean by the result of a measurement depends on whether the system is classical or quantum, but we should not be concerned with this distinction here. The only relevantpointisthatx carriesthemaximalamountofinformationonthesystemS t attimetwhichiscompatiblewiththelawsofphysics. In principle any physical system S that is not closed can be considered as part ofalargerbutclosedsystem.ItsufficestoconsiderwithS thesetRofallsystems which interact, in a way or another, with S. The joint system S ∨R is closed and from the knowledge of its state x at time t we can retrieve all the information on t itssubsystemS.InthiscasewesaythatthesystemS isopenandthatRisitsenvi- ronment.Therearehoweversomepracticalproblemswiththissimplepicture.Since the joint system S ∨R can be really big (e.g., the entire universe) it may be diffi- cult,ifnotimpossible,towritedownitsevolutionequation.Thereisnosolutionto VI Preface thisproblem.Thepragmaticwaytobypassitistoneglectpartsoftheenvironment R which, a priori, are supposed to be of negligible effect on the evolution of the subsystemS.Forexample,whendealingwiththemotionofachargedparticleitis oftenreasonabletoneglectallbuttheelectromagneticinteractionsandsupposethat theenvironmentconsistsmerelyintheelectromagneticfield.Moreover,iftheparti- clemovesinaverysparseenvironmentlikeintergalacticspacethenwecanconsider that it is the only source in the Maxwell equations which governs the evolution of R. Assuming that we can write down and solve the evolution equation of the joint systemS∨Rweneverthelesshitasecondproblem:howtochoosetheinitialconfig- urationoftheenvironment?IfRhasaverylarge(e.g.,infinite)numberofdegrees of freedom then it is practically impossible to determine its configuration at some initial time t = 0. Moreover, the dynamics of the joint system is very likely to be chaotic,i.e.,todisplaysomesortofinstabilityorsensitivedependenceontheinitial condition.Theslightesterrorintheinitialconfigurationwillberapidlyamplifiedand ruinourhopetopredictthestateofthesystematsomelatertime.Thus,insteadof specifyingasingleinitialconfigurationofRweshouldprovideastatisticalensem- ble of typical configurations. Accordingly, the best we can hope for is a statistical information on the state of our open system S at some later time t. The resulting theoryofopensystemsisintrinsicallyprobabilistic.Itcanbeconsideredasapartof statisticalmechanicsattheinterfacewiththeergodictheoryofstochasticprocesses anddynamicalsystems. TheparadigmofthisstatisticalapproachtoopensystemsisthetheoryofBrown- ianmotioninitiatedbyEinsteininoneofhiscelebrated1905papers[3](seealso[4] for further developments). An account on this theory can be found in almost any textbookonstatisticalmechanics(seeforexample[9]).Brownianmotionhadadeep impactnotonlyonphysicsbutalsoonmathematics,leadingtothedevelopmentof thetheoryofstochasticprocesses(seeforexample[12]). Open systems appeared quite early in the development of quantum mechanics. Indeed,toexplainthefinitelifetimeoftheexcitedstatesofanatomandtocompute the width of the corresponding spectral lines it is necessary to take into account the interaction of the electrons with the electromagnetic field. Einstein’s seminal paper [5] on atomic radiation theory can be considered as the first attempt to use a Markov process – or more precisely a master equation – to describe the dynamics of a quantum open system. The theory of master equations and its application to radiation theory and quantum statisticalmechanics was subsequently developed by Pauli[8],WignerandWeisskopf[13],andvanHove[11].Themathematicaltheory of the quantum Markov semigroups associated with these master equations started todevelopmorethan30yearslater,aftertheworksofDavies[2]andLindblad[7]. Itfurtherledtothedevelopmentofquantumstochasticprocesses. Toillustratethephilosophyofthemodernapproachtoopensystemsletuscon- siderasimple,classical,microscopicmodelofBrownianmotion.Eventhoughthis model is not realistic from a physical point of view it has the advantage of being exactly solvable. In fact such models are often used in the physics literature (see [10,6,1]). Preface VII BrownianMotion:ASimpleMicroscopicModel Inacubiccrystaldenotebyq thedeviationofanatomfromitsequilibriumposition x x∈Λ ={−N,...,N}3 ⊂Z3andbyp thecorrespondingmomentum.Suppose N x thattheinter-atomicforcesareharmonicandonlyactsbetweennearestneighborsof thecrystallattice.InappropriateunitstheHamiltonianofthecrystalisthen (cid:1) (cid:1) p2 κ x + xy(q −q )2, 2 4 x y x∈ΛN x,y∈Z3 where (cid:2) 1if|x−y|=1; κ = xy 0otherwise; andDirichletboundaryconditionsareimposedbysettingq = 0forx ∈ Z3\Λ . x N Iftheatomatsitex = 0isreplacedbyaheavyimpurityofmassM (cid:4) 1thenthe Hamiltonianbecomes (cid:1) (cid:1) p2 κ H ≡ x + xy(q −q )2, 2m 4 x y x∈ΛN x x,y∈Z3 where (cid:2) M ifx=0; m = x 1otherwise. We shall consider the heavy impurity at x = 0 as an open system S whose environmentRismadeofthe(2N+1)3−1remainingatomsofthecrystal.Towrite downtheequationofmotioninaconvenientformletusintroducesomenotation.We wseetΛde∗Nno=teΛbyNδ\x{th0e}(cid:3),Kqro=ne(cqkxe)rxd∈eΛl∗Nta,fpun=cti(opnx)axt∈xΛa∗Nn,dQby=|xq|0t,hPeE=ucpl0id.eFaonrnxor∈mZo3f x.Wealsosetχ= δ .ThemotionofthejointsystemS∨Risgovernedby |x|=1 x thefollowinglinearsystem q˙ =p, p˙ =−Ω2q+Qχ, 0 (1) MQ˙ =P, P˙ =−ω2Q+(χ,q), 0 where−Ω2 isthediscreteDirichletLaplacianonΛ∗ andω2 =6.Accordingtothe 0 N 0 opensystemphilosophydescribedinthepreviousparagraphweshouldsupplysome appropriatestatisticalensembleofinitialstatesoftheenvironment.Tomotivatethe choiceofthisensemblesupposethatintheremotepasttheimpuritywaspinnedat some fixed position, say Q = P = 0, and that at time t = 0 the resulting system hasreachedthermalequilibriumatsometemperatureT >0.Thepositionsandmo- mentainthecrystalwillbedistributedaccordingtotheGibbs-Boltzmanncanonical ensemblecorrespondingtothepinnedHamiltonianH =H| , 0 Q=P=0 (cid:4) (cid:5) 1 H = (p,p)+(q,Ω2q) . 0 2 0 VIII Preface ThisensembleisgivenbytheGaussianmeasure dµ=Z−1e−βH0(q,p)dqdp, whereZ isanormalizationfactorandβ =1/k T withk theBoltzmannconstant. B B Attimet = 0wereleasetheimpurity.Thesubsequentevolutionofthesystem isdeterminedbytheCauchyproblemforEqu.(1).Theevolutionoftheenvironment canbeexpressedbymeansoftheDuhamelformula (cid:6) sin(Ω t) t sin(Ω (t−s)) q(t)=cos(Ω t)q(0)+ 0 p(0)+ 0 χQ(s)ds. 0 Ω Ω 0 0 0 InsertingthisrelationintotheequationofmotionforQleadsto (cid:6) t MQ¨ =−ω2Q+ K(t−s)Q(s)ds+ξ(t), (2) 0 0 wheretheintegralkernelK isgivenby sin(Ω t) K(t)=(χ, 0 χ), (3) Ω 0 and (cid:7) (cid:8) sin(Ω t) ξ(t)= χ,cos(Ω t)q(0)+ 0 p(0) . 0 Ω 0 Sinceq(0),p(0)arejointlyGaussianrandomvariables,ξ(t)isaGaussianstochastic process.Itisasimpleexercisetocomputeitsmeanandcovariance 1 cos(Ω (t−s)) E(ξ(t))=0, E(ξ(t)ξ(s))=C(t−s)= (χ, 0 χ). (4) β Ω2 0 Wenoteinparticularthatthisprocessisstationary.Thetermξ(t)inEqu.(2)isthe noisegeneratedbythefluctuationsoftheenvironment.Itvanishesiftheenvironment isinitiallyatrest.TheintegralinEqu.(2)istheforceexertedbytheenvironmenton theimpurityinreactiontoitsmotion.Notethatthisdissipativetermisindependent ofthestateoftheenvironment.Thedissipativeandthefluctuatingforcesarerelated bythesocalledfluctuation-dissipationtheorem K(t)=−β∂ C(t). (5) t The solution zt = (Q(t),P(t)) of the random integro-differential equation (2) defines a family of stochastic processes indexed by the initial condition z0. These processesprovideastatisticaldescriptionofthemotionofouropensystem.Anin- variantmeasureρfortheprocessztisameasureonR3×R3suchthat (cid:6) (cid:6) f(zt)dρ(z0)= f(z)dρ(z), Preface IX holds for all reasonable functions f and all t ∈ R. Such a measure describes a steadystateofthesystem.Ifonecanshowthatforanyinitialdistributionρ which 0 isabsolutelycontinuouswithrespecttoLebesguemeasureonehas (cid:6) (cid:6) lim f(zt)dρ (z0)= f(z)dρ(z), (6) 0 t→∞ then the steady state ρ provides a good statistical description of the dynamics on largetimescales.Oneofthemainprobleminthetheoryofopensystemsistoshow thatsuchanaturalsteadystateexistsandtostudyitsproperties. TheHamiltonianApproach Remark that in our example, such a steady state fails to exist since the motion of thejointsystemisclearlyquasi-periodic.However,inarealsituationthenumberof atomsinthecrystalisverylarge,oftheorderofAvogadro’snumberN (cid:6)6·1023. A Inthiscasetherecurrencetimeofthesystembecomessolargethatitmakessenseto takethelimitN → ∞.Inthislimit−Ω2 becomesthediscreteDirichletLaplacian 0 ontheinfinitelatticeZ3\{0}.Thisisawelldefined,bounded,negativeoperatoron theHilbertspace(cid:11)2(Z3).Thus,Equ.(2),(3),(4)and(5)stillmakesenseinthislimit. Inthesequelweonlyconsidertheresultinginfinitesystem. Wedistinguishtwomainapproachestothestudyofopensystems.Thefirstone, theHamiltonianapproach,dealsdirectlywiththedynamicsofthejointsystemS∨R. Webrieflydiscussthesecondone,theMarkovianapproach,inthenextparagraph. IntheHamiltonianapproachwerewritetheequationofmotion(1)as Z˙ =−iΩ˜Z, whereΩ˜2 =m−1/2Ω2m−1/2withm=I+(M−1)δ (δ , ·)theoperatorofmulti- 0 0 plicationbym and−Ω2isthediscreteLaplacianonZ3.ThecomplexvariableZis x givenbyZ =Ω˜1/2m1/2q˜+iΩ˜−1/2m−1/2p˜andq˜=(qx)x∈Z3,p˜=(px)x∈Z3.Itfol- lowsfromelementaryspectralanalysisthatforM >1theoperator√Ω˜ isself-adjoint withpurelyabsolutelycontinuousspectrumσ(Ω˜)=σ (Ω˜)=[0, 2ω ]on(cid:11)2(Z3). ac 0 AsimpleargumentinvolvingthescatteringtheoryforthepairΩ2⊕ω2/M,Ω˜2shows 0 0 thatthesystemS hasauniquesteadystateρsuchthat(6)holdsforallρ whichare 0 absolutelycontinuouswithrespecttoLebesguemeasure.Moreover,ρisthemarginal onS oftheinfinitedimensionalGaussianmeasureZ−1e−βHdpdqdPdQwhichde- scribesthethermalequilibriumstateofthejointsystemattemperatureT =1/k β. B ThisiseasilycomputedtobetheGaussianmeasure ρ(dP,dQ)=N−1e−β(P2/2M+ω2Q2/2)dPdQ, whereN isanormalizationfactorand 1 ω2 = . (δ ,Ω−2δ ) 0 0 X Preface TheMarkovianApproach AremarkablefeatureofEqu.(2)isthememoryeffectinducedbythekernelK.Asa resulttheprocessztisnon-Markovian,i.e.,fors>0,zt+sdoesnotonlydependon zt and{ξ(u)|u ∈ [t,t+s]}butalsoonthefullhistory{zu|u ∈ [0,t]}.Theonly waytoavoidthiseffectistohaveKproportionaltothederivativeofadeltafunction. By Relation (5) thismeans that ξ should be a whitenoise. This iscertainly not the case with our choice of initial conditions. However, as we shall see, it is possible toobtainaMarkovprocessinsomeparticularscalinglimits.Thisisnotauniquely defined procedure: different scaling limits correspond to different physical regimes andleadtodistinctMarkovprocesses. Asasimpleillustrationletusconsidertheparticularscalinglimit Q (t)≡M1/4Q(M1/2t), M →∞. M ofourmodel.ForfiniteM theequationofmotionforQ reads M (cid:6) t Q¨ (t)=−ω2Q (t)+ K (t−s)Q (s)ds+ξ (t), M 0 M M M M 0 where K (t)≡M1/2K(M1/2t), M andthescaledprocessξ (t)≡M1/4ξ(M1/2t)hascovariance M C (t)≡M1/2C(M1/2t). M (cid:9) OnecanshowthatC(t)isinL1(R)andthatσ = C(t)dt > 0.Itfollowsthat,in distributionalsense, lim C (t)=σδ(t), lim K (t)=0. M M M→∞ M→∞ WeconcludethatthelimitingequationforQis Q¨(t)=−ω2Q(t)+σ1/2η(t), 0 whereη iswhitenoise,i.e.,E(η(t)η(s)) = δ(t−s).Thesolution(Q(t),Q˙(t))isa MarkovprocessonR3×R3withgenerator σ L=− ∆2 −P ·∇ +ω2Q·∇ . 2 P Q 0 P Itisasimpleexercisetoshowthattheuniqueinvariantmeasureofthisprocessisthe Lebesgue measure. Moreover, one can show that for any initial condition (Q ,P ) 0 0 andanyfunctionf ∈L1(R3×R3,dQdP)onehas (cid:6) lim E(f(Q(t),Q˙(t)))= f(Q,P)dQdP, t→∞ ascaledversionofreturntoequilibrium. Preface XI Itisworthpointingoutthatinmanyinstancesofclassicalorquantumopensys- tems the dynamics of the joint system S ∨R is too complicated to be controlled analytically or even numerically. Thus, the Hamiltonian approach is inefficient and the Markovian approximation becomes the only available option. The study of the Markovian dynamics of open systems is the main subject of the second volume in thisseries.Thethirdvolumeisdevotedtoapplicationsofthetechniquesintroduced inthefirsttwovolumes. Itaimsatleading thereader tothefrontof thecurrent re- searchonopenquantumsystems. OrganizationofthisVolume ThisfirstvolumeisdevotedtotheHamiltonianapproach.Itspurposeistodevelop the mathematical framework necessary to define and study the dynamics and ther- modynamicsofquantumsystemswithinfinitelymanydegreesoffreedom. ThefirsttwolecturesbyA.Joyeprovideaminimalbackgroundinoperatorthe- oryandstatisticalmechanics.ThethirdlecturebyS.Attalisanintroductiontothe theoryofoperatoralgebraswhichisthenaturalframeworkforquantummechanics ofmanydegreesoffreedom.Quantumdynamicalsystemsandtheirergodictheory are the main object of the fourth lecture by C.-A. Pillet. The fifth lecture by M. Merklidealswiththemostcommoninstancesofenvironmentsinquantumphysics, theidealBoseandFermigases.FinallythelastlecturebyV.Jaksˇic´ introducesone ofthemaintoolinthestudyofquantumdynamicalsystems:spectralanalysis. Lyon,GrenobleandToulon, Ste´phaneAttal September2005 AlainJoye Claude-AlainPillet

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Understanding dissipative dynamics of open quantum systems remains a challenge in mathematical physics. This problem is relevant in various areas of fundamental and applied physics. From a mathematical point of view, it involves a large body of knowledge. Significant progress in the understanding of
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