Repeated quantum interactions and unitary random walks

Stéphane Attal

Description

Niveau: Supérieur, Doctorat, Bac+8
Repeated Quantum Interactions and Unitary Random Walks Stephane ATTAL 1 and Ameur DHAHRI 2 1 Universite de Lyon, Universite de Lyon 1 Institut Camille Jordan, U.M.R. 5208 21 av Claude Bernard 69622 Villeubanne cedex, France 2 Facultad de Matematicas Campus San Joaquın Avenida Vicun˜a Mackenna 4860 Santiago, Chile Abstract Among the discrete evolution equations describing a quantum sys- tem HS undergoing repeated quantum interactions with a chain of exterior systems, we study and characterize those which are directed by classical random variables in RN . The characterization we obtain is entirely algebraical in terms of the unitary operator driving the el- ementary interaction. We show that the solutions of these equations are then random walks on the group U(H0) of unitary operators on H0. 1 Introduction In the article [AP], Attal and Pautrat have explored the Hamiltonian de- scription of a quantum system undergoing repeated interactions with a chain of quantum systems. They have shown that these “deterministic” dynamics give rise to quantum stochastic differential equations in the continuous limit. Since that result, some interest has been found in the repeated quantum in- teraction model in itself (cf [AJ1], [AJ2], [BJM1], [BJM2], [BP]) and several 1

• discrete evolution equations


• random walks


• repeated quantum


• time quantum


• undergoing repeated


• th interaction


• elementary operators


• quantum interaction


• quantum system

Sujets

• Jordan
• Quantum mechanics

Informations

• Publié par: mijec
• Nombre de lectures: 16
• Langue: English

Extrait

RepeatedQuantumInteractions dna UnitaryRandomWalks Ste´phaneATTAL 1 andAmeurDHAHRI 2 1 Universite´deLyon,Universite´deLyon1 InstitutCamilleJordan,U.M.R.5208 21avClaudeBernard 69622Villeubannecedex,France 2 FacultaddeMatema´ticas CampusSanJoaquı´n AvenidaVicun˜aMackenna4860 Santiago,Chile Abstract Amongthediscreteevolutionequationsdescribingaquantumsys- tem H S undergoingrepeatedquantuminteractionswithachainof exteriorsystems,westudyandcharacterizethosewhicharedirected byclassicalrandomvariablesin R N .Thecharacterizationweobtain isentirelyalgebraicalintermsoftheunitaryoperatordrivingtheel- ementaryinteraction.Weshowthatthesolutionsoftheseequations arethenrandomwalksonthegroupU( H 0 )ofunitaryoperatorson . H 0 1Introduction Inthearticle[AP],AttalandPautrathaveexploredtheHamiltoniande- scriptionofaquantumsystemundergoingrepeatedinteractionswithachain ofquantumsystems.Theyhaveshownthatthese“deterministic”dynamics giverisetoquantumstochasticdifferentialequationsinthecontinuouslimit. Sincethatresult,someinteresthasbeenfoundintherepeatedquantumin- teractionmodelinitself(cf[AJ1],[AJ2],[BJM1],[BJM2],[BP])andseveral 1

physicalworksareinprogressonthatsubject(forexample[KP]).These repeatedinteractionmodelsareinterestingforseveralreasons: –theyprovideaquantumdynamicwhichisatthesametimeHamiltonian andMarkovian, –theyallowtoimplementeasilythedissipationforaquantumsystem,in particulartheyarepracticalmodelsforsimulation, –theyexactlycorrespondtoactualphysicalsituations,inwhichaparticle, orafield,isundergoingrepeatedinteractionswithanothersystem(seee.g. [Har]), –theyareexactlythephysicalsituationsinwhichareperformedindirect measurementsofaquantumsystemandgiverisetotheso-called“quantum trajectories”(seee.g.[Pel]). TheprobabilisticnatureofthecontinuouslimitfoundbyAttalandPautrat isnotduetothepassagetothelimit,itisalreadybuilt-intheHamiltonian dynamicsofrepeatedquantuminteractions(itisactuallybuilt-intheaxioms ofquantummechanics). Theevolutionequationsdescribingtherepeatedquantuminteractionsare purelydeterministicbuttheyalreadyshowuptermswhichcanbeinterpreted as“discrete-timequantumnoises”.Thepointwiththesediscretequantum noisesisthatsometimestheymaygiverisetoclassicalnoises.Thatis,some linearcombinationsofthesequantumnoiseshappentobemutuallycommut- ingfamiliesofHermitianoperators,hencetheysimultaneouslydiagonalize andtheycanberepresentedasclassicalstochasticprocesses. Intheothercases,thatis,withdifferentcombinationsofthequantum noises,noclassicalprocessemergesandthedynamicsofrepeatedquantum interactionsarepurelyquantum. Theaimofthearticleistocharacterisealgebraically,ontheHamiltonian, thecasewhenthedynamicsareclassicallydriven. Thearticleisorganisedasfollows.Wefirst(Section2)presentthephys- icalandmathematicalsetupsfortherepeatedquantuminteractions.In Section3weintroducethebasicalgebraictool:theobtuserandomwalks whichareanappropriate“basis”ofrandomwalksadaptedtothislanguage. Wethenexploreandcharacterisetheunitaryrandomwalkswhichemerge classicallyfromtherepeatedquantuminteractions(Section4).Wespecial- izeinSection5ourresulttotheonedimensionalcasewhichalreadyshows upanon-trivialstructure.Finally,thelastsectionisdevotedtophysical examples;weexhibitexplicitHamiltoniansgivingrisetoclassicaldynamics. 2