Quantum phase transitions in an effective Hamiltonian fast a

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

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