Quantum phase transitions in an effective Hamiltonian fast and slow systems

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

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 "

2

Y,ωX ωY,andtheinteractionconstantgsatis esthestrongcouplingcondition:ωX g ωY.Weshowthat,dependingonthetypeofinteractionandthenatureofquantumsystemsdi erentphysicalsituationstakeplace,butgenericallysuche ectiveHamiltoniansdescribeQuantumPhaseTransitionsintheslowsystem.

II.

EFFECTIVEHAMILTONIAN

Now,wewillbeinterestedinthelimitwheretheslowsystemfrequencyislessthan/oroftheorderofthecou-plingconstant,ω1 g ω2.FollowingthemethoddescribedinRef.[2]wecanadiabaticallyremoveallthetermsthatcontainthefastsystem’stransitionop-erators,Y±.Inparticular,thecounter-rotatingtermX+Y++X Y andtherotatingtermX+Y +X Y+canbeeliminatedfromtheHamiltonian(1)byasubsequentapplicationofthefollowingLie-typetransformations:

U1=exp[ε(X+Y+ X Y )],U2=exp[ (X+Y X+Y )],

(4a)(4b)

LetusconsiderthefollowinggenericHamiltoniande-scribinganinteractionbetweentwoquantumsystems:

H=ω1X0+ω2Y0+g(X++X )(Y++Y ),

(1)

whereX0andY0arethefreeHamiltoniansoftheXandYsystemsrespectively,andsuchthatω1 ω2.TheaboveHamiltoniandoesnotpreservethetotalexcita-tionnumberoperatorN=X0+Y0and,inthelimitω1,ω2 g,leadstotheappearanceofmultiphoton-nm

typeinteractionsoftheformX+Y which,undercer-tainphysicalconditionsonthefrequenciesω1,2,describeresonanttransitionsbetweenenergylevelsofthewholesystem(see[3]andreferencestherein).

Theraising-loweringoperatorsX±,Y±describetran-sitionsbetweenenergylevelsofthesystemsXandYrespectivelyandconsequentlyobeythefollowingcom-mutationrelations:

[X0,X±]=±X±,

[Y0,Y±]=±Y±.

(2)

wherethesmallparameters,εand ,arede nedby 1.(5)

ω2 ω1

Thetransformations(4a)and(4b)generatedi erent

nknk

kindsofterms:suchasX±Y±+h.c.,X±Y +h.c.,nn

Y±+h.c.,andX±+h.c.withcoe cientsdependingonX0andY0.Undertheconditionω1,g ω2alltherapidlyoscillatingterms,i.e.thosecontainingpowersofY±,canberemovedbyapplyingtransformationssimilarto(4),withproperlychosenparameters.Then,theef-fectiveHamiltonianisdiagonalfortheoperatorsoftheYsystem.Theresultcanbeexpressedasapowerseriesofthesingleparameterδ=g/ω2 1.ε=

Itisworthnotingthatitisnotenoughthatδbeasmallparameterfortheformalexpansionin(4)(andthesubsequenttransformations).Abalanceisnecessarybe-tweenthee ectivedimensionsofthesubsystemsandδ.Thee ectivedimensionsofthesystemdependontheorderofthepolynomialsφ1,2,andonthepowersoftheelementsX±,0andY±,0involvedineachtransformation.Itwasshownbefore[3],thatthepowersofthesmallparametersareincreasingfasterthanthepowersofX±,0andY±,0,whichimpliesthatwecanfocusonthee ectivedimensionsintroducedwith(4).

Takingintoaccounttheabovementionedconsidera-tions,keepingonlytermsuptothirdorderinδanddis-regardingsmallcorrectionstothee ectivetransitionfre-quencies,wearriveatthefollowinge ectiveHamiltonian:

g

Wedonotimposeanyconditiononthecommutatorsbetweentransitionoperators,whicharegenerallysomefunctionsofdiagonaloperatorsandofsomeintegralsofmotion[N1,X0]=[N2,Y0]=0:

[X+,X ]= X0φ1(X0,N1),[Y+,Y ]= Y0φ2(Y0,N2),

(3a)(3b)

whereφ1(X0,N1)=X+X andφ2(Y0,N2)=Y+Y aresomepolynomialsofX0andY0respectively(fromnowonweomitthedependenceonintegralsN1,2inthearguments)and zφ(z)=φ(z) φ(z+1).Theobjects(X0,X±)and(Y0,Y±)areknownaspolynomialdeformedalgebrasslpd(2,R)[11].

Heff=ω1X0+ω2Y0 2ω1δ2 x, yΦ(X0,Y0+1)+gδ yφy(Y0)(X++X )

1+

2

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 "

Becausethee ectiveHamiltonian(6)isdiagonalfor

the

operators

of

theY

fastsystem,wemayprojectitoutontoaminimalenergyeigenstateoftheYsystem,|ψ0 Y,substitutingY0byitseigenvaluey0:Y0|ψ0 Y=y0|ψ0 Y.The rstordere ectthencomesfrom(X++X )2the,whiletheterm～(X++X )4

term～

de nesa nestructureofthee ectivepotential,obtainedafterprojectingthee ectiveHamiltonian(6)ontothestate|ψ0 Y.

Itisimportanttostressthat,althoughδisasmallparameter,thee ectoftheterms～δn,n≥1,couldbeinprinciplecomparablewiththemaindiagonaltermω1X0,especiallyifthealgebraofXoperatorsdescribeabigsubsystem,i.e.,largespinorbigphotonnumber.Inthiscasenon-triviale ectssuchasQPTmayoccur.Now,wemayproceedwithanalysisofthee ectiveHamiltonian(6)inthethermodynamiclimit,focusingonthepossiblebifurcationofthegroundstate.

III.

EXAMPLES

A.

Atom- eldinteraction(Dickemodel)

TheHamiltoniangoverningtheevolutionofAsym-metricallypreparedtwo-levelatomsinteractingwithasinglemodeofquantized eldhastheform

H=ω1n +ω2Sz+g(S++S ) a +a

,(8)wheren =a aandSz,±aregeneratorsofthe(A+1)-dimensionalrepresentationofthesu(2)algebra.

1.E ective elddynamics

Firstletussupposethattheatomsformafastsubsys-temsothat,

X0=n ,X+=a ,X =a,

Y0=Sz,Y±=S±,

andthus,φy(Y0)=C2 S2

Cz+Sz

andφx(X0)=n ,where

2=A/2(A/2+1)istheeigenvalueoftheCasimiropera-torofthesu(2)algebra(integralofmotioncorrespondingtotheatomicsubsystem).

Projectingthee ectiveHamiltonianontothemini-mumenergystateoftheatomicsystem|0 at,sothaty0= A/2,weobtainthefollowinge ectiveHamilto-nianforthe eldmode:

Heff=ω 1n Agδ

a+a 2+gAδ3 a+a 4,(9)whereω 1=ω1(1 2Aδ2).

Rewriting(9)intermsofpositionandmomentumop-erators,

Heff=

ω 1

3

2

cosθ,Sx→

A

2

sinφsinθ,

andthusrewritethee ectiveHamiltonian(11)asaclas-sicalHamiltonianfunction,

Hcl=

A

1 ξ 2for

ξ>1.ItisworthnotingthattheglobalminimumofHclatξ<1convertsintoalocalmaximumforξ>1,sothatHcl(θ )<Hcl(θ ).Thismeansthattheatoms,initiallypreparedattheminimumoftheHamiltonianfunction,spontaneouslychangetheirgroundstateenergyatsomevalueofthesystem’sparameters.Classically,

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 "

thisimpliesappearanceofaseparatrix,whichleadstothediscontinuityontheenergydensityspectruminthethermodynamiclimit.Itisalsoworthnotingthatthereisalossoftherotationalsymmetryinthisprocess:thenewgroundstateisobviouslynotinvariantunderrotationsaroundaxisx,whiletheinitialgroundstateisclearlyinvariantunderx-rotations.

Itiseasytoseethatthelasttwotermsin(12)areoflowerorderintheparameterAδandcanbeneglectedinthe rstapproximationfordescriptionofQPTatξ=1.

B.

Spin-spininteraction

ingsystemwhenthetotalexcitationnumberisnotpre-served.Analyzingthosee ectiveHamiltoniansinthethermodynamiclimitwehaveobservedabifurcationofthegroundstateleadingtothee ectoftheQuantumPhaseTransitions.

Itisinterestingtonotethat,formultidimensionalsys-tems,whenalgebraicallytheXsystemisadirectsumofseveralnon-interactingsubsystems,aninterestinge ectofgenerationofentangledstates(inthenon-preservingexcitationcase)canbeobserved.Really,letussupposethatin(1)X0,±=X0,±1+X0,±2,[Xj,1,Xj,2]=0,j=0,±;thenthecorrespondinge ectiveHamiltonian(uptoa rstnon-trivialorderinδ)takestheformHeff≈ω1(X0,1+X0,2)+ω2Y0+

gδ[(X+,1+X ,1)2+(X+,2+X ,2)2

+2(X+,1+X ,1)(X+,2+X ,2)] yφy(Y0),

wherewecanclearlyseethatthelasttermcontainstheoperatorproduct～X+,1X+,2which,togetherwithquadratictermsinX±,1(2),impliesaspontaneousgen-erationofentangledstatesofX1andX2startingfromtheminimumenergystate.Thiscanbecorroboratedbytheentanglingpowermeasurebyconsideringauniformdistributionoftheinitialfactorizedstates[12].Thuswecansaythat,intheregimestudiedhere,entanglementcanbegeneratedinabipartitesystemthevicinityofaPhaseTransition.

Asasecondexampleletusconsideradipole-dipolelikeinteraction,thatis,

X0=Sz1,X±=S±1,

Y0=Sz2,Y±=S±2.

Thee ectiveHamiltonianfortheslowspinsystem(afterprojectingontotheloweststateofthefastspinsystemwitheigenvalue A2/2)takestheformsimilarto(10),with

222

Heff=ω 1Sz1 2A2gδSx1+2A2ω1δSz1

4322

+16gA2δ3Sx1+24gδA2{Sx,Sz},

(13)

whereω 1=ω1 2A1ω1δ2 20gδ3A21.The rsttwo

termsaredominantforδ 1anddescribetheLipkin-Meshkovmodel,sothatthecriticalpointisreachedatξ=4A2A2 1=1inthethermodynamicallimit.The1gδ/ω

e ectoftherestofthetermsin(13)isnegligibleinthevicinityofξ=1.

IV.

CONCLUSIONS

Acknowledgments

Wededucethee ectiveHamiltonianofagenericslowquantumsysteminteractingwithanotherfastoscillat-

Oneoftheauthors(I.S.)thanksSTINT(SwedishFoundationforInternationalCooperationinResearchandHigherEducation)forsupport.Thisworkwassup-portedbyGrants:CONACyTNo

1080535.

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