Berry Phase and Fidelity in the Dicke model with A2 term Yu-Yu Zhang1,2, Tao Liu3, Qing-Hu Chen2,1,†, Kelin Wang3,4 1 Department of Physics, Zhejiang University, Hangzhou 310027, P. R. China. 2 Center for Statistical and Theoretical Condensed Matter Physics, Zhejiang Normal University, Jinhua 321004, P. R. China 3 Department of Physics, Southwest University of Science and Technology, Mianyang 621010, P. R. China. 4 Department of Mordern Physics, University of Science and Technology of China, Hefei 230026, P. R. China. (Dated: January 18, 2009) The instability, so-called the quantum-phase-like transition, in the Dicke model with a rotating- wave approximation for finiteN atoms is investigated in terms of the Berry phase and thefidelity. It can be marked by the discontinuous behavior of these quantities as a function of the atom-field 9 coupling parameter. Involvingan additional field A2 term, it is observed that the instability is not 0 eliminated beyond the characteristic atom-field coupling parameter even for strong interaction of 0 thebosonic fields, contrarily to theprevious studies. 2 PACSnumbers: 42.50.Nn,64.70.Tg,03.65.Ud n a J I. INTRODUCTION terms of the BP and the GS fidelity. 8 In this paper, we calculate the ground state BP and 1 fidelity in the RWA DM with and without A2 term to TheDickemodel(DM)[1]describesanensembleartifi- ] cialtwo-levelatomscouplingwithacavitydevice. Ithas quantify phenomena of the quantum-phase-like transi- h tions in finite system. Without A2, a exact solution to been attracted considerable attentions recently, mainly p the DM is given explicitly. We solve the RWA DM with - due to the fact that the Dicke model is closely related to nt manyrecentinterestingfieldsinquantumopticsandcon- an additional A2 term by a exact diagonalization in the Fock space of the bosonic operators. The paper is or- a densedmatterphysics,suchasthesuperradiantbehavior u by an ensemble of quantum dots [2] and Bose-Einstein ganized as follows. In Sec.II, we review the RWA DM q condensates[3],coupledarraysofopticalcavitiesusedto and the Hamiltonian with the A2 term to obtain exact [ solutionsrespectively. InSec.III,westudytheinstability simulate and study the behavior of strongly correlated of the RWA DM by measuring the BP and the ground 1 systems[4], and superconducting charge qubits[5, 6]. It state fidelity. The behaviors of these two quantities as v is known from the previous studies[7, 8, 9] that the full 5 DM undergoes the second-order quantum phase transi- a function of the interaction strength of the field for the 0 RWA DM with A2 term are also evaluated. Finally, we tion [10]. 7 present the conclusion in Sec.IV. As claimed in Ref. [11], a sequence of instabilities, 2 . so-called quantum-phase-like transitions, is involved in 1 the problem of an ensemble of two-level atoms system 0 II. MODEL interacting with a bosonic field in the rotating-wave ap- 9 proximation (RWA)[12, 13, 14]. As addressed Ref. [15], 0 : the absence of field A2 term from the minimal coupling A. Exact solution to the RWA DM v Hamiltonian in the approximation of the DM leads to i X the possibility of the instability. In the presence of A2 Let us consider DM of N two-level atoms with energy r term, the classical thermodynamic properties have been levelω0,interactingwithasingle-modebosonicfieldwith a studied previously [16, 17, 18]. Whether the instabilities the frequency ω. In the RWA DM, ignoring the counter- disappear when the interaction of the bosonic field A2 rotating term, the corresponding Hamiltonian is given term is taken into account is a long-standing issue and by remains very controversialto date[12, 15, 16]. It is knownthat quantum critical phenomena exhibits H =ωa†a+ω0Jz + λ (a†J−+aJ+). (1) deep relations to the Berry phase (BP) [19, 20, 21, 22] √N and the fidelity [23, 24, 25, 26, 27, 28, 29, 30]. The BP has been extensively studied by the geometric time wherea†andaarethephotoniccreationandannihilation evolution of a quantum system, providing means to de- operators, Jk(k = z, ) denotes the collective spin-1/2 ± tect the quantum effects and critical behavior, such as atomic operators, λ is the atom-field coupling strength, quantum jumps and collapse [31, 32, 33]. A recent pro- and h¯ is set unity. posal is to use the fidelity in identifying the quantum Motivated by the exact technique of the Jaynes- phase transition[24, 25]. As a consequence of the dra- Cammings model with RWA, we present a detailed nu- matic changes in the structure of the ground states, the mericaldiagonalizationprocedurestosolveasetofclosed fidelityshoulddropatcriticalpoints. Inouropinion,the equations to the DM with RWA, which was also briefly quantum-phase-like transitions might also be studied in discussed in Ref. [12]. Since the Hamiltonian ( 1) 2 commutes with the total excitation number operator Lˆ =a†a+J + 1, the subspace ofthe Hilbert space con- z 2 sistsofasumofsubspaceslabeledbydifferentnumberof excitationsL. TheHilbertspaceofthecollectivealgebra N=1 10 N=2 is spanned by the kets j,m ;m = j, j +1, ,j . {| i − − ··· } N=4 By adapting Schwinger’s representation of spin in terms 8 N=8 of harmonic oscillators [34, 35], j,m can be expressed | i as j,N n , which is a Dike state of N n spin-up 6 ato|ms an−d ni spin-down atoms, n = 0,1,..−.N. In this L work, j takes its maximal value N/2. N/2,N n 4 is also known as the eigenstates of J |and J2 −withi z Jz|N2,N−ni=(N2 −n)|N2,N−ni. Theactionofthecor- 2 responding raising and lowering operators on this state gives 0 0 0.5 1.0 1.5 2.0 N N J+ ,N n = (N n+1)n ,N n+1 | 2 − i − | 2 − i p FIG. 1: Excitation number L versus atom-field coupling pa- N N rameter λ for different numbersof atoms N =1,2,4,8. J ,N n = (N n)(n+1) ,N n 1 − | 2 − i − | 2 − − i p . step by step and keeps a constant in a coupling parame- In the subspace of L = N + k excitations the wave terinterval[λ ,λ ],whereλ (λ )isaquantum-phase-like function is supposed as i j i j transition point, as shown in Fig. 1. The first transi- N tion point is denoted as λ0c. The sensitive quantities like ψ = c n+k N/2,N n the groundstateBPandthe fidelitywillbe calculatesto | i n| if | − ia nX=0 O quantify the discontinuities, so called instability, in the (k = N, N +1,...) (2) finite DM with RWA. − − where c ’s are coefficients, n+k is a Fockstate ofthe n f | i bosonic field with an alterative number k, ranging from B. Numerical exact diagonalization to the RWA N to infinity. k is equal to N in the weak coupling DM with A2 term − − regionscorrespondingto0excitationsandthenincreases with the coupling parameter λ. When the number of It is interesting to discuss the effect of the interacting excitations L is larger than the number of atoms N, i.e. bosonic field in the atom-field system. As the interac- k >0,thegroundstateofH liesinthesubspacespanned tions vector potential A, causedby the longitudinal part by N +1 vectors. In this way, the Hamiltonian is ex- of the bosonic field, are taken into account, the Hamil- pressedbyatridiagonal(N+k+1) (N+k+1)matrix. × tonian in the RWA DM can be evaluated with an addi- FromEq.( 2) we obtain the exactexpressionofthe m-th tional term A2 [16, 17].The additional term A2 has been row of the Schr¨odinger equation discussedclassicallyaboutthermodynamicpropertiesby λ Rza¸z˙ewski et al [17, 18]. To extensively quantify the Ecm = √m+k (N m+1)mcm−1 contributions of the A2 term, we employ the quantum √N − p information tools such as the BP and the ground state λ + √m+k+1 (N m)(m+1)cm+1 fidelity to detect the quantum-phase-like transitions. √N p − Intermsofthebosonicoperators,theA2 termisgiven +[ω(m+k)+ω0(N/2 m)]cm (3) by ε(a†+a)2, where ε is the interacting strength of the − bosonic field. The overall Hamiltonian of the ensemble where of two-level atoms interacting with the bosonic field is 0,1,...N+k : k 0 expressed as m= ≤ (4) (cid:26) 0,1,...N : k >0 λ Note that the above equations are closed and the set HA =ωa†a+ω0Jz+√N(a†J−+aJ+)]+ε(a†+a)2 (5) of linearequations for c′s takes a tridiagonalform. Solu- tionsforagivenk arereadilyobtainedthroughGaussian In order to obtain the numerical exact solution, we per- elimination and back substitution. Finally the chosen k formastandardBogoliubovtransformationby introduc- corresponds to the lowest energy among eigenvalues of ing bosonic annihilation (creation) operator b(b†), such the solutions for a fixed coupling strength λ. It is in- thatb† =µa+νa† and µ2 ν 2 =1. Aftersubstituting teresting to find that the excitation number L is added a, a† into Eq.( 5) the t|ot|al−H|a|miltonian is diagonalized 3 as HA = ω2+4ωεb†b+ω0Jz + 1( ω2+4ωε ω) 2 − 0.0 p p N=1 λ N=2 + [µ(b†J−+bJ+) ν(b†J++bJ−)] (6) -0.5 N=4 √N − N=64 -1.0 where N -0.7 1 ω+2ε 1 ω+2ε -1.5 -0.8 N=64 µ2 = ( +1),ν2 = ( 1) -0.9 2 √ω2+4ωε 2 √ω2+4ωε − -2.0 N -1.0 -1.1 Notethatacounter-rotatingtermisincludedinthemod- -2.5 -1.2 -1.3 ified Hamiltonian ( 6), which may plays an essentialrole -1.4 in the following discussion. Because the A2 term breaks -3.0 1.5 1.6 1.7 1.8 1.9 2.0 the gauge invariance of the Hamiltonian( 1) in the DM 0 0.5 1.0 1.5 2.0 2.5 3.0 with RWA, it was argued in Ref. [15] the instability would be then eliminated. WenowconsiderthewavefunctionsofthetotalHamil- FIG. 2: The average Berry phase γ1/N of the RWA DM as tonian with N atoms, which are of the form a function of the coupling parameter λ for different numbers of atoms N = 1,2,4,64. The inset shows a discontinuous N Ntr pictureof instability for 64 atoms. ϕ = d m N/2,n (7) A nm f a | i | i | i nX=0mX=0 where Ntr is the maximum photonic number in the ar- Therefore,A2termintheDMmodelwithoutRWAwould tificially truncated Fock space, and dnm are coefficients. notchangethenatureofthephasetransition,exceptthat m f isaFockstatewithmphotons. N/2,n aisaDicke the position of the critical point is shifted. | i | i state in Schwinger’s representation of spin with n atoms in excited state. The m-th row of the Schr¨odinger equa- tion reads III. GROUND STATE PROPERTY N 1 Ed = [ω m+∆(n )+ (ω ω)]d nm ε ε nm − 2 2 − A. Instability in the RWA DM λµ + (m+1)(N n+1)ndn−1,m+1 √N − Berry’s adiabatic geometric phase describes a phase p λµ factorofthewavefunctionsinatime-dependentquantum + m(n+1)(N n)dn+1,m−1 system. Theinterestingpathsofevolutionforgenerating √N − p a BP are those for which the ground state of the system λν (m+1)(n+1)(N n)dn+1,m+1 can evolve around a region of criticality. We first mea- − √Np − sure a nontrivial BP circulating a region of ”criticality” λν m(N n+1)ndn−1,m−1 (8) correspondingto a abruptchange. The BP γ1 generated − √N − after the system undergoing the time-dependent unitary p transformation U(T)=exp[ iφ(t)J ], varying the angle The eigenvalues and eigenfunctions can be obtained nu- − z φ(t) adiabatically from 0 to 2π, can be evaluated as a merically by diagonalizinga (N+1) (Ntr+1) matrix. × function of the atom-field coupling parameter λ The BP and the fidelity can be calculated through these eigenfunctions. 2π d Tobecomplete,wealsobrieflyreviewthecontribution γ1 =i ψ0 U†(t) U(t)ψ0 dφ=2π ψ0 Jz ψ0 of the A2 term in the DM model without RWA, which Z0 h | dφ | i h | | i (11) yields some unimportant corrections. The Hamiltonian of the full DM with the A2 term reads where |ψ0i is the ground-state wave function of Hamil- tonian (1) of the RWA DM. As shown in Fig. 2, the λ HDM =ωa†a+ω0Jz+ √N(a†+a)Jx+ε(a†+a)2 (9) fithrestcpohuapslein-tgrapnasriatimonet-leirkeλo0c=cu1rsfaotrtahrebi”tcrrairtyicaalt”omvalnuuemo-f c With a rotation around an y axis by an angle π and the ber N, which recover the result in the thermodynamical same Bogoliubov transformation, the modified2Hamilto- limit [36]. The average BP γ1/N is equal to π at λ < 1 and increases abruptly at discontinuous ”critical” points nian H is rewritten as DM when λ >1. Note that the plateau is formed clearly for HDM = ω2+4ωεb†b ω0Jx+ 1( ω2+4ωε ω) N = 1,2,4 , the width of the plateau becomes narrower − 2 − andnarrowerwith the increasingN, whicharequite dif- p p 2λ ferentfrom the phenomenonofthe quantum phase tran- + (µ ν)(b†+b)J . (10) √N − z sition in the full DM [7, 8, 9]. A clear picture of the 4 1.0 1.0 2.0 0.9 N=64 0.8 F N0.7 0.5 1.5 0.6 N=1 N 0.5 N=2 N=4 1.0 0.4 0 1.5 1.6 1.7 1.8 1.9 2.0 0.4 0.5 N=1 N=2 0.2 N=4 0.0 N=64 0.0 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 c FIG. 3: The average Berry phase γ2/N.i.e. the average FIG.4: GroundstatefidelityF andenergygap∆betweenthe phononnumber,inunitsof2π oftheRWADMasafunction energiesofthefirstexcitedandgroundstateintheRWADM oTfhteheincsoeutpgliinvegspaardaimsceotnetrinλufooursdpifficetruenretnoufminbsetraboifliattyomfosrN64. as a function of the coupling parameter of λ/λ0c for different numberof atoms N =1,2,4,∞. atoms. instability in the ground state of the RWA DM is given turbation parameter, i.e. intermsoftheBPwithN =64atomsshownintheinset of Fig. 2. F(λ,δλ)= ψ0(λ)ψ0(λ+δλ) (13) |h | i| The effect of decoherence of the driving field on adia- batic evolutionsofspinandquantizedmodes systemhas NotethatF isafunctionofbothλandδλ. Basedonthe been investigated[20,21]. Inthe fully quantizedcontext normalized and orthogonalized wave function in Eq.(2) we need a procedure capable of generating an analogous fortheRWADM,thegroundstatefidelitycanbederived tpahianseedcihnatnegrme isnofththeestbaotseonofictohpeefiraeltdo.rbTyhtehBePphγa2seisshoibft- aannadlythtiecnalclyan|bPeNns,imm=p0licfine(dλ)acsm(λ + δλ)δn,mδn+k,m+k′|, unitary operator R(φ)=exp[ iφ(t)nˆ], where nˆ =a†a is − the number of bosons in the field. Changing the angle 0 : n=m,k =k′ φ(t) slowly from 0 to 2π the ground state γ2 is given by F(λ,δλ)=(cid:26)1 : n=m,k =6 k′ (14) 2π d γ2 =i ψ0 R†(t) R(t)ψ0 dφ=2π ψ0 a†aψ0 At each transition point in RWA DM, the alternative Z0 h | dφ | i h | | i number k is changed abruptly and F is then equal to 0. (12) Beyond ”critical” points, F should be constant 1. The We now have a general expression for the BP γ2 re- energy gap∆ between the firstexcited andgroundstate lated to the photonic number,which is driven by fields. energies tends to 0 at each ”critical” point, as shown We plot behaviors of γ2/N in units of 2π as a function in Fig. 4. We attribute this level crossing to the fact oftheatom-fieldcouplingparameterλfordifferentnum- that the GS wave functions at the different sides of each ber of atoms N in Fig. 3. The BP γ2/N is 0 in the transitionpointareorthogonal. Thegroundstatefidelity weak coupling region for λ 1 and then increases dis- ≤ dropsto0ateach”critical”point,exhibitingsimilarcrit- continuously as λ increases. As shown in Fig. 3, when ical singularity of the first-order quantum phase transi- N increasestheintervalofthe“critical”valuesλbecome tion [28]. It is also obvious that the instabilities increase smaller, leading to the curve with more steps. The inset with a increasing number of atoms. of Fig. 3 shows that there actually exist many phase- transition-like “critical” points beyond λ = 1 for large N =64. An increasing interest has been drawn in the role of B. Behaviors of the Berry phase and fidelity in the the groundstate fidelity in detecting the quantum phase RWA DM with A2 term transitionsforvariousmany-bodysystem,withanarrow dropatthetransitionpoint. Belowweproposetousethis We next turn to study the RWA DM with A2 term by quantum tool to identify the quantum-phase-like transi- usingtheabovequantuminformationtools. Bymanseof tions, where the GS fidelity drops to 0 in the RWA DM. the general BP formula Eqs.( 11) and ( 12), the ground It is defined as the overlap between two ground states state BP of the RWA DM with A2 term can be evalu- ψ0(λ) and ψ0(λ+δλ) [26, 27], where δλ is a tiny per- ated as a function of the atom-field coupling λ and the | i | i 5 1 FA0.5 -0.5 0 -1.0 ∂λ ∂γ/1A 3 ε=1 N-1.5 00 / 1A ε=0.0 -2.0 −1 ε=10−3 N γ/1A−2 εε==1100−−21 -2.5 ε=100 N=4 ω=ω=1 −3 0 -3.0 (a) 0 0.5 λ/1λ 1.5 2 0 0.5 1.0 1.5 2.0 c FIG.6: GroundstatefidelityFA (a),Berryphaseγ1A/N (b) and its first derivative ∂γ1A/∂λ (c) of the RWA DM with A2 term as a function of the coupling of λ/λc for different interacting strengthes of thefield ε=0,10−3,10−2,10−1,1. 0.8 0.6 N /2A 0.4 coupling parameter λ where the first abrupt jump oc- curs as a characteristic parameter λ . It is interesting c 0.2 to observe that the characteristic λ increases with the c interactionstrength ε. The stability claimed in Ref. [15] 0.0 (b) is only found in the smooth curves for λ λ . Thus, c 0 0.5 1.0 1.5 2.0 there still exist quantum-phase-liketransitio≤ns when the interaction of the bosonic field ε is strong. Accordingto the groundstatewavefunction ϕ , the A fidelity of the RWA DM with A2 term canbe al|so cialcu- 1.0 lated F (λ,δλ) = ϕ (λ)ϕ (λ+δλ) . The numerical A A A |h | i| resultsforthedifferentεwithN =4atomsareexhibited inFig. 5(c). The singularitiesatthe ”critical”points for FA0.5 ε = 0,0.5,1 are demonstrated by a sudden drop of FA. It is clearly shown that the characteristic λ moves to- c wards the right regime with the increasing ε, providing the evidence of the quantum-phase-like transitions even for a strong interaction of the field. 0.0 (c) 0 0.5 1.0 1.5 2.0 To show the instabilities more obviously, for a more wide range of interacting strengthes ε = 0,10−3,10−2,10−1,1 , we plot the ground state fidelity FIG. 5: The average Berry phase (a) γ1A/N, (b) γ2A/N in FA, Berry phase γ1A/N and its first derivative ∂γ1A/∂λ unitsof2π,aswellas(c)groundstatefidelityFAoftheRWA ofthe RWADMwithA2 termasafunctionofthescaled DMwithA2termversusλfordifferentinteractingstrengthes coupling parameter λ/λ together in Fig. 6. When the of the field ε=0,0.5,1 for N =4. c interaction strength of the field increases the fidelity F still drops to 0 at the characteristic λ , where the first c interaction strength of the field ε, i. e., derivative N−1∂γ1A/∂λ also changes abruptly. This is anotherpieceofevidencethattheinstabilityoftheRWA γ1A =2π ϕA Jz ϕA , DM does not vanish in the RWA DM including the A2 h | | i γ2A =2π ϕA b†bϕA (15) term. h | | i where ϕ is the ground state of the Hamiltonian (5). It is illustrated that the contribution of the A2 term A | i WeplottheaverageBPγ1A/N inFig. 5(a)andγ2A/N does not eliminates the instability of the RWA DM, con- inunitsof2π inFig. 5(b)asafunctionofthe couplingλ trarily to the previous studies by Rza¸z˙ewski et al.[15]. fordifferentinteractionstrengthesε=0,0.5,1forN =4 For strong interaction strength of the bosonic field ε, a atoms. Wecanobservethat,forbothγ1A/N andγ2A/N, characteristic parameter λc becomes larger than λ0c = 1 the abrupt jump occurs at the same coupling parameter intheabsenceofA2term. Asequenceofthegroundstate λ for the same interaction strengthes ε. We denote the stability reported previously only appears for λ λ . c ≤ 6 IV. CONCLUSIONS quantities to detect quantum phase (or like) transition. Insummary,wehaveinvestigatedtheinstabilityofthe RWA DM by quantum information tools such as the BP V. ACKNOWLEDGEMENTS and the ground state fidelity. An obvious discontinuous behaviorsofthese quantities withfinite N atomsareob- This work was supported by National Natural Science served. It is demonstrated that the quantum-phase-like Foundation of China, PCSIRT (Grant No. IRT0754) in transitions occur beyond the characteristicλ for strong UniversityinChina,NationalBasicResearchProgramof c interaction of fields. We propose that the instability China (Grant No. 2009CB929104), Zhejiang Provincial would not be eliminated by involving the A2 term of the Natural Science Foundation under Grant No. Z7080203, DMwithRWA.Previousobservedinstabilitymaybelim- and Program for Innovative Research Team in Zhejiang ited to the coupling regime λ λ . It should be pointed Normal University. c outthatthequantuminformat≤iontoolsareverysensitive † Corresponding author. Email:[email protected] [1] R.H. Dicke,Phys. Rev. 93, 99(1954). Phys. Rev.Lett. 92, 020402(2004). [2] M. Scheibneret al., NaturePhys.3, 106(2007). [22] F. Plastina, G. Liberti, and A. Carollo, Europhys. Lett. [3] D.Schnebleet al., Science 300, 475 (2003). 76,182(2006). [4] M.J.Hartmannetal.,NaturePhys.2,849(2006); A.D. [23] H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Greentree et al., ibid. 2, 856(2006). Sun,Phys. Rev.Lett. 96, 140604 (2006). [5] J. Q. You,and F.Nori, Phys.Today. 58(11), 42(2005). [24] W. L. You, Y. W. Li, and S. J. Gu, Phys. Rev. B. 76, [6] A.M.Zagoskinetal.,Phys.Rev.Lett.97,077001(2006). 022101 (2007); S. J. Gu, H. M. Kwok, W. Q. Ning, and [7] K.Hepp and E. Lieb, Ann.Phys., 76, 360(1973). H. Q. Lin, ibid.77, 245109(2007). [8] C. Emary and T. Brandes, Phys. Rev. E 67, [25] M. Cozzini, R. Ionicioiu, and P. Zanardi, Phys. Rev. B 066203(2003); Phys. Rev.Lett. 90, 044101(2003). 76, 104420 (2007). [9] Q.H.Chen,Y.Y.Zhang,T.Liu,andK.L.Wang,Phys. [26] M. F. Yang,Phys. Rev.B. 76, 180403 (2007). Rev.A. 78, 051801(R)(2008). [27] S.Chen,L.Wang,S.J.Gu,andY.Wang,Phys.Rev.E. [10] S. Sachdev, Quantum Phase Transitions (Cambridge 76, 061108 (2007). UniversityPress, Cambridge, England, 2000). [28] S. Chen, L. Wang, Y. J. Hao, and Y. P. Wang, Phys. [11] L.M.Narduccietal.,CollectivePhenomena1,113(1973); Rev. A 77, 032111 (2008). Phys.Rev.A. 8, 1892(1973). [29] H. Q. Zhou, R. Orus, G. Vidal, Phys. Rev. Lett. 100, [12] V. Buˇzek, M. Orszag, and M. Roˇsko, Phys. Rev. Lett. 080601 (2008). 94, 163601(2005). [30] T.Liu,Y.Y.Zhang,Q.H.Chen,andK.L.Wang,arXiv: [13] B. V. Thompson, J. Phys. A. 10, 89(1977). 0812.0321. [14] B. V. Thompson, J. Phys. A. 10, L179(1977). [31] S. Dusuel, and J. Vidal, Phys. Rev. Lett. 93, [15] K.Rza¸z˙ewski, and K.W´odkiewicz, Phys. Rev.Lett. 96, 237204(2004). 089301 (2006). [32] A.C.M.Carollo, andJ.K.Pachos,Phys.Rev.Lett.95, [16] K. Rza¸z˙ewski, and K. W´odkiewicz, Phys. Rev. A. 43, 157203(2005). 593 (1991). [33] S. J. Zhu,Phys. Rev.Lett. 96, 077206(2006). [17] K. Rza¸z˙ewski, K. W´odkiewicz, and W. Zakowicz, Phys. [34] J. Schwinger, L. C. Biedenharn, and H. Vandam, In Rev.Lett. 35, 432(1975). Quantumtheoryofangularmomentum,NewYork: Aca- [18] K. Rza¸z˙ewski, and K. W´odkiewicz, Phys. Rev. A. 13, demic, pp.229-279 (1965). 1967(1976). [35] J.J.Sakurai,Modernquantummechanics,Reading,MA: [19] M.V.Berry,Proc.R.Soc.London,Ser.A,392,45(1984). Addison-Wesley,(1994). [20] I. F. Guridi, A. Carollo, S. Bose, and V. Vedral, Phys. [36] G. Chen, J. Q. Li, and J. Q. Liang, Phys. Rev. A. 74, Rev.Lett. 89, 220404(2002). 054101(2006). [21] A. Carollo, I. F. Guridi, M. F. Santos, and V. Vedral,