An effective Hamiltonian describing interaction between generic "fast" and a "slow" systems is obtained in the strong interaction limit. The result is applied for studying the effect of quantum phase transition as a bifurcation of the ground state of the "
Quantumphasetransitionsinane ectiveHamiltonian:fastandslowsystems
IsabelSainz,1A.B.Klimov,2andLuisRoa3
SchoolofInformationandCommunicationTechnology,
RoyalInstituteofTechnology(KTH),Electrum229,SE-16440Kista,Sweden
2
DepartamentodeF´ sica,UniversidaddeGuadalajara,Revoluci´on1500,44420Guadalajara,Jalisco,Mexico.
3
CenterforQuantumOpticsandQuantumInformation,DepartamentodeF´ sica,
UniversidaddeConcepci´on,Casilla160-C,Concepci´on,Chile.
(Dated:February3,2008)
Ane ectiveHamiltoniandescribinginteractionbetweengenericfastandaslowsystemsisob-tainedinthestronginteractionlimit.Theresultisappliedforstudyingthee ectofquantumphasetransitionasabifurcationofthegroundstateoftheslowsubsysteminthethermodynamiclimit.Examplesasatom- eldandatom-atominteractionsareanalyzedindetail.
PACSnumbers:42.50.Ct,42.50.Hz,42.50.Fx
1
arXiv:0801.4689v1 [quant-ph] 30 Jan 2008
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I.INTRODUCTION
Frequently,intheprocessofinteractionbetweentwoquantumsystems,onlyoneofthemcanbedetectedex-perimentally.Inthiscase,avarietyofphysicale ectsappearintheprocessofsuchinteractionwhichcanbede-scribedintermsofane ectiveHamiltoniancorrespond-ingtotheobservedsystem.Thesimplestexampleofsuchasituationariseswhenafastsysteminteractswithaslowsystem.Then,thefastsystemcanbeadiabaticallyelim-inatedandtheslowsystemisdescribedbyane ectiveHamiltonian.TheseconsiderationswereassumedinthefamousBorn-Oppenheimerapproximation.Aregularap-proachtothequantumdynamicsoftheobservedsystemisprovidedbytheLietransformationmethodTheadvantageofthismethodconsistsinthepossibilityofvaryingthesystem’sparameters,changingrelationsbe-tweenthem,whichallowsustodescribedi erentphysicalregimesusingthesamemathematicaltool.Inparticu-lar,suchanimportantexampleasexpansionontheres-onancesinquantumsystemsnotpreservingthenumberofexcitationscanbeobtainedInthiscaseagenericHamiltoniangoverninginteractionoftwosubsystemsbe-yondtheRotatingWaveApproximation(RWA)canberepresentedasaseriesinoperatorsdescribingallpossibletransitionsinthesystem.
Severalinterestingfeaturesappearingintheprocessofinteractionofquantumsystemscanberealizedbystudy-ingevolutionofonlytwogenericquantumsystemwithonequantumchannel.Eveninsuchasimplecasewemaydiscriminateatleastthreeinterestinglimits:a)whentheinteractionconstantgismuchhigherthanthecharacter-isticfrequenciesofbothinteractingsystems;b)whengissmallerthanthefrequenciesofthesystemsandc)whengishigherthanthefrequencyofonesystembutsmallerthatthefrequencyoftheotherone.
Thea)caseofverystrongcouplingshouldbestudiedcarefully,becauseusingtheexpansionparameterlikeaninteractionconstantoveracharacteristicfrequencycouldbequitetricky.Forinstance,thetypeofthespectrumcorrespondingtothenon-perturbedandtotheperturbed
systemscanbedi erent:eithercontinuousordiscrete.Theb)casecorrespondstoasituationwherethereso-nanceexpansionisapplicable.Thisparticularcaseleadstodispersive-likeinteractionsAsitwasshowninRef.theevolutionisgovernedbyane ectiveHamiltoniandescribingacertainresonantinteractionandtherepre-sentationspaceofthetotalsystemcanbealwaysdividedinto(almost)invariantsubspaces.
Thelastcasec)possessesapeculiarproperty:besidesof ndingacorrespondinge ectiveHamiltonian,wecanalsoprojectitouttothelowerenergystateofthefastsystem,whichwouldnevergetexcitedundergivenre-lationsbetweenthesystem’sparameters,andthus,de-scribeane ectivedynamicsoftheslowsysteminthelimitofstronginteraction.Itiswellknownthatinthisregimesuchaninterestinge ectasQuantumPhaseTran-sitionsmayoccur.
Thequantumphasetransitions(QPT)areacommonfeatureofnon-linearquantumsystems.Suchtransitionsoccuratzerotemperatureandareassociatedwithanabruptchangeinthegroundstatestructure.QPTarerelatedtosingularitiesintheenergyspectrumand,atthecriticalpointsde ningQPT,thegroundstateen-ergyisanon-analyticfunctionofthesystem’sparame-tersQualitatively,forawideclassofquantumsys-tems,severalimportantpropertiesofQPTcanbestud-iedintheso-calledthermodynamic(semiclassical)limit[7,8].Then,QPTcanbeanalyzedintermsofaclassicale ectivepotentialenergysurface[9].InthislanguageQPTarerelatedtotheappearanceofanewclassicalseparatrixwhenthecouplingparametersacquirecertainvalues.Accordingtothestandardsemiclassicalquantiza-tionschemeandthecorrespondenceprinciple,theenergydensityisproportionaltotheclassicalperiodofmotion,divergingontheseparatrix,whichexplainsahighdensityofquantumstatesatthecriticalpoints.
Inthisarticlewestudye ectiveHamiltoniansdescrib-ingevolutionofagenericquantumsystemXinteract-ingwithaquantumsystemYinthecasewherethecharacteristicfrequencyofthesystemXisessentiallylowerthanthecorrespondingfrequencyofthesystem