Diffractive energy spreading and its semiclassical limit

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1、DiffractiveenergyspreadinganditssemiclassicallimitAlexanderStotlandandDoronCohenDepartmentofPhysics,Ben-GurionUniversity,Beer-Sheva84005,IsraelAbstract.Weconsiderdrivensystemswherethedrivinginducesjumpsinenergyspace:(1)particlespulsedbyasteppotential;(2)particlesinaboxwithamovingwall;(3)p

2、articlesinaringdrivenbyanelectro-motive-force.Inallthesecasestheroutetowardsquantum-classicalcorrespondenceishighlynon-trivial.Someinsightisgainedbyobservingthatthedynamicsinenergyspace,wherenisthelevelindex,isessentiallythesameasthatofBlochelectronsinatightbindingmodel,wherenisthesitein

3、dex.Themeanlevelspacingislikeaconstantelectricfieldandthedrivinginduceslongrangehopping∝1/(n−m).1.IntroductionConsiderasystemwhichisdescribedbyaHamiltonianH(X(t)),wheretheparameterX(t)istimedependent.ForsuchsystemtheenergyEisnotaconstantofthemotion.Rather,thedrivinginducesspreadinginenerg

4、yspace.Assumingthatthesystemispreparedatt=0inamicrocanonicalstate,onewondershowtheenergydistributionρt(E)lookslikeatalatertime.Inparticular,onemaywonderwhetherthequantumρt(E)issimilartothecorrespondingclassicaldistribution.Inthe“quantumchaos”literatureitiscustomarytodistinguishbetweenacl

5、assicaltimescaleτclandaquantumbreaktimet∗.Thelattergoestoinfinityinthe“~→0”limit.Aprototypemodelisthe“quantumkickedrotator”[1]wheretheenergyspreadingisdiffusiveuptot∗whileforlargertimesoneobservessaturationduetoadynamicallocalizationeffect.Inthisworkweanalyzemuchsimplersystemswherethebreakt

6、imet∗,ifexists,ismuchlargerthananyphysicallyrelevanttimescale.Infactonemayassumethatthetimetoftheevolutioniscomparablewiththeclassical(short)timescale.Insuchcircumstancesonenaivelywouldexpectquantumtoclassicalcorrespondence(QCC).Butinfactthetheoryismuchmorecomplicated[2].Onehastodistingu

7、ishbetween•DetailedQCC•RestrictedQCCDetailedQCCmeansthatallthemomentsr=1,2,3,...ofthequantummechanicaldistributionρt(E)aresimilartotheclassicalresult,whilerestrictedQCCrefersonlyarXiv:cond-mat/0605591v2[cond-mat.mes-hall]10Aug2006tother=1,2moments.Itturnsoutthatthelattera