Quantum phase transitions in an effective Hamiltonian fast a(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 "

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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+

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