Stochastic Master Equations in Thermal Environment

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Niveau: Supérieur, Doctorat, Bac+8
Stochastic Master Equations in Thermal Environment Stephane ATTAL Universite de Lyon, Universite de Lyon 1 Institut Camille Jordan, U.M.R. 5208 21 av Claude Bernard, 69622 Villeurbanne cedex, France Clement PELLEGRINI ? Institut de Mathematiques de Toulouse Laboratoire de Statistique et de Probabilite Universite Paul Sabatier (Toulouse III) 118 Route de Narbonne, 31062 Toulouse Cedex 9, France (Received 2008) Abstract. We derive the stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. At zero temperature it is well-known that one obtains stochastic differential equations of jump-diffusion type. At strictly positive temperature, we show that only pure diffusion type equations are relevant. 1. Introduction The theory of Open Quantum Systems aims to study the time evolution of a small system H0 interacting with an environment E , cf [1, 2, 24, 26]. Starting from an Hamiltonian de- scription of the coupled system [2, 24, 26], the evolution of the reduced system H0 is obtained by tracing over the degree of freedom of the environment.

  • been presented

  • thermal gibbs

  • open quantum

  • has been

  • master equations

  • quantum repeated

  • indirect quantum measurements

  • equation

  • quantum stochastic


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StochasticMasterEquationsinThermalEnvironmentSte´phaneATTALUniversite´deLyon,Universite´deLyon1InstitutCamilleJordan,U.M.R.520821avClaudeBernard,69622Villeurbannecedex,Franceattal@math.univ-lyon1.frCle´mentPELLEGRINIInstitutdeMathe´matiquesdeToulouseLaboratoiredeStatistiqueetdeProbabilite´Universite´PaulSabatier(ToulouseIII)118RoutedeNarbonne,31062ToulouseCedex9,Franceclement.pellegrini@math.univ-toulouse.fr(Received2008)Abstract.Wederivethestochasticmasterequationswhichdescribetheevolutionofopenquantumsystemsincontactwithaheatbathandundergoingindirectmeasurements.Theseequationsareobtainedasalimitofaquantumrepeatedmeasurementmodelwhereweconsiderasmallsystemincontactwithaninfinitechainatpositivetemperature.Atzerotemperatureitiswell-knownthatoneobtainsstochasticdifferentialequationsofjump-diffusiontype.Atstrictlypositivetemperature,weshowthatonlypurediffusiontypeequationsarerelevant.1.IntroductionThetheoryofOpenQuantumSystemsaimstostudythetimeevolutionofasmallsystemH0interactingwithanenvironmentE,cf[1,2,24,26].StartingfromanHamiltoniande-scriptionofthecoupledsystem[2,24,26],theevolutionofthereducedsystemH0isobtainedbytracingoverthedegreeoffreedomoftheenvironment.IntheMarkovianapproachofopensystems,thetimeevolutionofthestateofthereducedsystemischaracterizedbyasemigroupofcompletelypositivemaps,withatypicalgeneratorcalledLindbladgenerator,whichgivesrisetoanordinarydifferentialequationcalledmasterequation[2,24,28].Inthisframework,anactivelineofresearch,motivatedbyrecentexperimentalapplicationsinquantumopticsandquantumcommunications,isfocusedonthedescriptionofquantummeasurement[7,8,9,10,11,12,13,14,24,27,28,26].Basically,inordertoavoidZenoeffect[24],themeasurementisperformedontheenvironment.Accordingtothepostulatesofquantummechanics,thisinvolvesarandomperturbationoftheevolutionofthestateofH0.ThedynamicsofH0isthendescribedbyclassicalstochasticdifferentialequations,whichareperturbationsofthemasterequationintermsofwhitenoise[7,22,23,8,9,10,11,12,21,30,32,37,38].Usually,theseequationsarecalledStochasticSchro¨dingerEquationsorStochasticMasterEquationsandtheirsolutionsarecalledQuantumTrajectories(thename“stochasticSchro¨dingerequation”isusuallyreservedfortheevolutionofthestateofH0intermsofpurestateswhereasstochasticmasterequationsconcernsevolutionofdensitymatrices).WorksupportedbyANRproject“HAM-MARK”NANR-09-BLAN-0098-01.ExemplaryOSIDstyle