Egorov property in perturbed cat map

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1、EgorovpropertyinperturbedcatmapMartinHorvat(1,2),MirkoDegliEsposti(2)(1)PhysicsDepartment,FacultyofMathematicsandPhysics,UniversityofLjubljana,Slovenia(2)DepartmentofMathematics,UniversityofBologna,ItalyE-mail:martin.horvat@fmf.uni-lj.si,desposti@dm.unibo.itAbstract.Westudythetimeevo

2、lutionofthequantum-classicalcorrespondence(QCC)forthewellknownmodelofquantisedperturbedcatmapsonthetorusintheveryspecificregimeofsemi-classicallysmallperturbations.ThequalityoftheQCCismeasuredbytheoverlapofclassicalphase-spacedensityandcorrespondingWignerfunctionofthequantumsystemcall

3、edquantum-classicalfidelity(QCF).IntheanalysedregimetheQCFstronglydeviatesfromtheknowngeneralbehaviourdiscussedin[1],inparticularitdecaysfasterthenexponential.HerewestudyandexplaintheobservedbehavioroftheQCFandtheapparentviolationoftheQCCprinciple.Submittedto:J.Phys.A:Math.Gen.1.Intro

4、ductionThequantum-classicalcorrespondence(QCC)isthebasicprincipleunderlyinganyphysicalquantisationofaclassicalsystem.AccordingtothisprinciplethequantumsystemshouldbehavesimilartothecorrespondingclassicalsystemwithincreasingarXiv:0706.4397v2[quant-ph]16Jul2007energyordecreasingeffectiv

5、ePlanckconstant.TheimportanceofQCCasatoolinthestudyofquantumsystemswasrecognisedveryearlyinthedevelopmentofquantummechanicswiththeEhrenfesttheoremandlaterbytheintroductionofsemi-classicalmethods[2].ThestudyofQCCgaveinthe80’sbirthtoquantumchaology–researchareadevotedtostudytheconnecti

6、onsbetweendynamicalpropertiesofclassicalsystemsandcorrespondingquantumsystems[3].TheQCCcanbeexploredanddiscussedusingvarioustoolsandmethodsavailableinthetheoryofclassical/quantumsystems.Inarecentpaper[1],aphasespacerepresentationhasbeenusedtostudythetimeevolutionofQCCingenericchaotic

7、systemsoncompactclassicalphasespace.TheQCCistherequantifiedusingthesocalledquantum-classicalfidelity(QCF),namelytheintegratedoverlapbetweentheclassicalphase-spacedensityandthecorrespondingWignerfunction.Ithasbeenshownthatinclassicallychaoticsystems,aftersomeinitialplateau,theQCFdecayse

8、xponentially