Quantum phase transitions in an effective Hamiltonian fast a(3)

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,