010 Zeg Ke et al Vol. 14 level j+i ad a lower level j i. The upper level j+i is coupled to the lower level j i by the sigle-mode vacuum field. I the i

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1 Vol 14 No 10, October 005 cfl 005 Chi. Phys. Soc /005/14(10)/ Chiese Physics ad IOP Publishig Ltd Quatum etaglemet i the system of two two-level atoms iteractig with a sigle-mode vacuum field * Zeg Ke(Ξ ) a)b) ad Fag Mao-Fa( Λ ) a)y a) Departmet of Physics, Hua Normal Uiversity, Chagsha , Chia b) Departmet of Applied Physics ad Electroic Techology, Chagsha Uiversity, Chagsha 41000, Chia (Received 5 April 005; revised mauscript received 8 Jue 005) The etaglemet properties of the system of two two-level atoms iteractig with a sigle-mode vacuum field are explored. The quatum etaglemet betwee two two-level atoms ad a sigle-mode vacuum field is ivestigated by usig the quatum reduced etropy; the quatum etaglemet betwee two two-level atoms, ad that betwee a sigle two-level atom ad a sigle-mode vacuum field are studied i terms of the quatum relative etropy. The iflueces of the atomic dipole dipole iteractio o the quatum etaglemet of the system are also discussed. Our results show that three etagled states of two atoms field, atom atom, ad atom field ca be prepared via two two-level atoms iteractig with a sigle-mode vacuum field. Keywords: quatum etaglemet, sigle mode vacuum field, reduced etropy, relative etropy PACC: Itroductio Quatum etaglemet (QE) is oe of the most strikig properties of quatum mechaics. It is a key problem i Eiste Podolsky Rose (EPR) Paradox, [1] Bell's iequalities, [] quatum cryptography, [] quatum teleportatio, [4] quatum computatio [5] ad so o, ad has bee viewed as a elemetary resource for quatum iformatio processig. It is essetial to create ad maipulate etagled states for quatum iformatio applicatio. Most experimetal works of quatum etaglemet have bee implemeted with photos. Although idividual polarizatio states of photos are easily cotrolled ad their quatum coherece ca be preserved over may kilometers i a optical fibre, [6] photos caot be stored for a log time, ad the maipulatio of collective etagled states demostrates cosiderable difficulties eve if photos are cofied i the same cavity. The creatio of log lived etagled pairs with atoms, o the other had, is a relatively recet pursuit, which may provide reliable quatum iformatio storage. The etagled state of a pair of two-level atoms by usig pulse area techique i a microwave cavity has bee realized by Hagley [7] based o the proposal of Cirac ad Zoller. [8] The geeratio of atomphoto etaglemet has also bee proposed i Ref.[9] i a tripod-like laser-atom-cavity system, which sustais two cavity modes. I this paper, we will explore the geeratio of atomic ad atom field etagled states via the system of two two-level atoms iteractig with a siglemode vacuum field. We will ivestigate the degree of the etaglemet (DE) betwee two two-level atoms ad a sigle-mode vacuum field by usig the quatum reduced etropy. I additio we will also study the DE betwee two two-level atoms, ad that betwee a sigle two-level atom ad a sigle-mode vacuum field through usig the quatum relative etropy. [10].Reduced desity matrices of two two-level atoms ad a sigle-mode vacuum field Λ Project supported by the Natioal Natural Sciece Foudatio of Chia (Grat No ). y Correspodig author. mffag@huu.edu.c The system cosidered here cosists of two twolevel atoms iteractig with a sigle-mode vacuum field. We assume a two-level atom with a upper

2 010 Zeg Ke et al Vol. 14 level j+i ad a lower level j i. The upper level j+i is coupled to the lower level j i by the sigle-mode vacuum field. I the iteractig picture, the effective Hamiltoia [11] i the rotatig wave approximatio (RWA) [1;1] is (μh = 1) with ad H I = H 0 =! 0 H = H 0 + H I ; (1) S (i) +!a + a; () g(a + S (i) + as (i) + ) + Ω(S (1) + S() + S (1) S() + ): () Here a + ad a are the creatio ad aihilatio operators of the field mode of frequecy!;! 0 is the atomic trasitio frequecy; S, S + ad S are the usual pseudo-spi operators of the two-level atom; g is the atom-field couplig costat, ad Ω is the atomic dipole-dipole couplig costat, i(= 1; ) deotes ith atom. For simplicity, we cosider oly the resoat case (! =! 0 ). We cosider that at t = 0 the two atoms are i the excited statej++i ad the field is i a sigle-mode vacuum state j0i. The iitial state of the system is a decoupled pure state, ad the state vector is jψ(0)i = jψ a1a (0)i ΩjΨ f (0)i = j ++0i: (4) I the iteractig picture, at ay time t > 0 the evolutio of the state vector of the system obeys the Schrödiger equatio (μh = jψ I(t)i = H I jψ I (t)i: (5) Itca be obtaied by solvig the Schrödiger equatio jψ I (t)i =C 0 (t)j ++0i + C 1 (t)j + 1i where the coefficiets [11] are + C (t)j +1i + C (t)j i; (6) C 0 (t) = g A + ; C 1 (t) = C (t) = g B; C (t) = p g A p ; (7) with A = eiat a eibt b ; B = e iat e ibt ; = p Ω +4g ; a = ( Ω p + Ω +4g ) ; b = ( Ω p Ω +4g ) : (8) The desity matrix of the system is ρ(t) = jψ I (t)ihψ I (t)j: (9) The reduced desity matrix of the subsystem composed of two atoms is ρ a1a (t) = tr f (jψ I (t)ihψ I (t)j) = 6 4 C 0 C Λ C 1 C Λ 1 C 1 C Λ 0 0 C C Λ 1 C C Λ C C Λ Its eigevalues ca be obtaied as follows, 1 (t) = C 0 C Λ 0 ; (t) = C 1 C Λ 1 + C C Λ ; (t) = C C Λ ; 7 5 : (10) 4 (t) = 0: (11) Similarly, we ca obtai the reduced desity matrix of the sigle-mode vacuum field ρ f (t) =tr a1a ρ(t) =C 0 C Λ 0 j0ih0j + C 1C Λ 1 j1ih1j + C C Λ j1ih1j + C C Λ jihj = 1 (t)j0ih0j + (t)j1ih1j + (t)jihj: (1) It turs out that the eigevalues of ρ f (t) are idetical to those of ρ a1a (t). Accordig to the Schmidt theorem, [14] the state vector of the system ca be rewritte as jψ I (t)i = p 1 (t)j ++ij0i + p (t)=j + ij1i + p (t)=j +ij1i + p (t)j iji: (1) I what follows, we will utilize the above results to ivestigate the etaglemet properties of the system ad the subsystems.

3 No. 10 Quatum etaglemet i the system of two two-level atoms Quatum etaglemet betwee two two-level atoms ad a sigle-mode vacuum field We use the reduced quatum etropy as the measure of the DE betwee two two-level atoms ad a sigle-mode vacuum field. The etropies of the subsystem ca be defied through their respective reduced-desity matrix as [15] are disetagled from the vacuum field, while at time t = ( 1)ß=(4g), S a1a (t) evolves to its maximal values, the two atoms are strogly etagled with the vacuum field, which meas that the two atoms-field etagled state ca be prepared by choosig the time t = ( 1)ß=(4g). S i (t) = tr[ρ i (t)lρ i (t)]; (i = a 1 a ;f): (14) The etropies of a geeral two-compoet quatum system are liked to a remarkable theorem preseted by Araki ad Lieb, [16] which states that js a1a (t) S f (t)j»s a1a f(t)»js a1a (t)+s f (t)j: (15) Here S a1af (t) = tr[ρ(t)lρ(t)] is the total etropy of the system of two two-level atoms ad the siglemode vacuum field. It is worth otig that the ρ(t) give by Eq.(9) is govered by a uitary time evolutio ad cosequetly the total etropy S a1af (t) is time idepedet. Sice we have assumed that the two two-level atoms ad the sigle-mode vacuum field are iitially i a disetagled pure state, the total etropy S a1af(t) of the system is zero. Oe immediate cosequece of iequality (15) is S a1a (t) = S f (t). As a result, we oly eed to calculate the atomic quatum etropy S a1a (t). We ca express the atomic quatum etropy i terms of the eigevalues i (t) of the atomic reduced desity matrix give by Eq.(10) S a1a (t) = i (t)l i (t): (16) It reflects the DE betwee the two two-level atoms ad the sigle-mode vacuum field: if S a1a (t) takes its miimal value zero, the two atoms ad the field are disetagled. If S a1a (t) takes its ozero value, the two atoms ad the field are etagled. Figure 1 displays the umerical results for the time evolutio of the reduced etropy S a1a (t) for k = Ω=g = 0, 10, 0. I Fig.1 (a), k=0 correspodig to the time evolutio of the reduced etropy S a1a (t) i the absece of the dipole-dipole iteractio betwee the two two-level atoms. It is observed that the reduced etropy S a1a (t) evolves with a period of ß. Whe t = ( 1)ß=g, ( = 1; ; ;:::), S a1a (t) evolves to its zero values ad the two two-level atoms Fig.1. The time evolutio of the reduced etropy S a1 a(t). (a) k = 0; (b) k = 10; (c) k = 0. The results for the time evolutio of reduced etropy S a1a (t) i the presece of the dipole dipole iteractio are plotted i Figs.1 (b) ad (c). From these figures, we ca see the iflueces of the atomic dipole-dipole iteractio o the atomic reduced etropy S a1a (t). It is observed that the dipole-dipole iteractio of the two atoms leads to the decrease of the maximal values of the atomic reduced etropy ad the icrease of its time evolutio period, which meas that i order to prepare the two atoms-field etagled state, the ifluece of the atomic dipole dipole iteractio should be reduced. 4. Quatum etaglemet betwee two two-level atoms The above results show that the effective Hamiltoia give by Eqs.() ad () will lead to the etaglemet betwee two two-level atoms ad the siglemode vacuum field. Therefore, the states of atomatom ad a sigle atom-field may evolve ito mixed states. I these cases, the reduced etropy caot measure the DE betwee atom-atom ad that betwee a sigle atom field. However, the relative etropy of the etaglemet is a good measure for the DE of twoatom mixed state ad the DE of a sigle atom field, which is defied as [17] E R = mi S(ρjjff); (17) ffd

4 01 Zeg Ke et al Vol. 14 where S(ρjjff) = tr[ρ(l ρ l ff)] is the quatum relative etropy, its miimum is take over D, the set of all disetagled states. The relative etropy of the etaglemet is viewed as the miimal `distace' betwee the state ρ ad the disetagled state ff. For a pure state, the relative etropy of the etaglemet reduces to its reduced etropy, while for mixed states, it is usually difficult to calculate the relative etropy of the etaglemet except for some specific states. Recetly, the followig theorem about the relative etropy of the etaglemet has bee prove, [18;19] ad it is cosiderably suitable for our aalysis. If a bipartite quatum state ca be expressed as ρ = 1; a 1; jψ 1 ffi 1 ihψ ffi j: (18) The the relative etropy of the etaglemet is give by E j (ρ j ) = a ; l a ; + tr(ρ j l ρ j ) (j = a 1 a ;af); (19) ad the disetagled state ff that miimizes the quatum relative etropy S(ρjjff) is give by ff = a ; jψ ffi ihψ ffi j; (0) where jψ i ad jffi i are orthoormal states of each subsystem. It should be oted that the reduced desity matrix give by Eq.(10) could take the form ρ a1a = ;m a ;m j; ihm; mj; (1) where a ;m = h; jρ a1a jm; mi. Hece, the relative etropy of etaglemet for the state i Eq.(10) is writte as E a1a (ρ a1a ) = = + a ; l a ; S(ρ a1a ) a ; l a ; i (t)l i (t): () Therefore, E a1a (t) reflects the DE betwee two two-level atoms. The umerical results of Eq.() are show i Fig. ad the parameters used are the same as i Fig.1. It is obvious that the relative etropy E a1a (t) also evolves with a regular periodicity. I Fig. (a), we assume that the atomic dipole-dipole iteractio is abset. At time t = ( 1)ß=(5g), ( = 1; ; ; ) the relative etropy achieves its miimum, which meas that the two two-level atoms are disetagled. Fig.. The time evolutio of the relative etropy E a1 a(t). (a) k = 0; (b) k = 10; (c) k = 0. At time t = ( 1)ß=(5g), the relative etropy reaches its maximum, i.e. the two two-level atoms are strogly etagled. This result shows that at time t = ( 1)ß=(5g), the atom-atom etagled state ca be prepared. The results for the time evolutio of relative etropy of the two atoms i the presece of the dipole-dipole iteractio are show i Figs.(b) ad (c). It is show that the atomic dipole-dipole iteractio leads to the decrease of the maximal values of the atomic relative etropy ad the shorteig of its time evolutio period. 5. Quatum etaglemet betwee a sigle two-level atom ad a sigle-mode vacuum field By tracig a atom for the desity matrix of the system, the reduced desity matrix of a sigle twolevel atom ad a sigle-mode vacuum field is obtaied i the atom-field bases j +0i, j 1i, j +1i ad j i: ρ af (t) = tr a ρ(t) = 6 4 C 0 C Λ 0 C 0 C Λ C 1 C Λ 0 C 1 C Λ C C Λ C C 7 Λ 5 : () 0 0 C C Λ C C Λ

5 No. 10 Quatum etaglemet i the system of two two-level atoms Its eigevalues r j (t), (j=1,,, 4) are give by r 1 (t) = C 0 C Λ 0 + C 1 C Λ 1 ; r (t) = C C Λ + C C Λ ; r (t) = r 4 (t) = 0: (4) The relative etropy of the sigle atom-field is E af (t) = i=0 C i C Λ i l(c i C Λ i )+ j=1 r j (t)lr j (t): (5) The time evolutio of E af (t) reflects the time evolutio of the DE betwee a sigle two-level atom ad a sigle-mode vacuum field. E af (t) is plotted i Fig.. The parameters are the same as i Fig.1. Fig.. The time evolutio of the relative etropy E af (t). (a) k = 0; (b) k = 10; (c) k = 0. It is iterestig to ote that the etaglemet dyamics of a sigle atom ad a sigle-mode vacuum field is almost the same as that of the two atoms i sectio 4. This result shows that at time t = ( 1)ß=(5g), the atom-field etagled state ca be prepared. 6. Coclusio I this paper, we explore the quatum etaglemet of the system of two two-level atoms iteractig with a sigle-mode vacuum field. We study the DE betwee two two-level atoms ad a sigle-mode vacuum field by usig the quatum reduced etropy, ad ivestigate the DE betwee the two two-level atoms, ad that betwee a sigle two-level atom ad a siglemode vacuum field by usig quatum relative etropy. We ca coclude as follows: First, three etagled states (two atoms-field; atom-atom ad atom-field) ca be prepared via two two-level atoms iteractig with a sigle-mode vacuum field. Secod, three kids of the QE evolve periodically with the same phase. Fially, the atomic dipole dipole iteractio leads to the decrease of the degrees of the etaglemet. I order to prepare etagled states, the ifluece of the atomic dipole dipole iteractio should be reduced. Our results are importat for the experimetal realizatio of the preparatio of etagled states. Refereces [1] Eistei A, Podolsky B ad Rose N 195 Phys. Rev [] Bell J S 1964 Physics [] Ekert A K 1991 Phys. Rev. Lett Scully M O ad Zubairy M S 1997 Quatum Optics (Cambridge: Cambridge Uiversity Press) p51 [4] Beett C H et al 199 Phys. Rev. Lett [5] Cirac J I ad Zoller P 1995 Phys. Rev. Lett Wu Y et al 004 Phys. Rev. A [6] Giovaetti V, Lloyd S ad Maccoe L 001 Nature Giovaetti V, Lloyd S ad Maccoe L 00 Phys. Rev. A [7] Hagley E et al 1997 Phys. Rev. Lett [8] Cirac J I ad Zoller 1994 Phys. Rev. A 50 R799 [9] Ba M 1999 J. Opt. B 1 L9 [10] Nielse M A ad Chuag I L 000 Quatum Computatio ad Quatum Iformatio (Cambridge: Cambridge Uiversity Press) p504 [11] Peg J S ad Li G 1996 Itroductio of Moder Quatum Optics (Beijig: Sciece Press) p [1] Sukumar C V ad Buck B 1981 Phys. Lett. A 8 11 Wu Y ad Yag 1997 Phys. Rev. Lett [1] Cavers C M ad Schumaker B L 1985 Phys. Rev. A [14] Phoeix S J D ad Kight P L 1988 A. Phys. (N Y) Schmidt E 1906 Math. A. 6 4 [15] Wehrl A 1978 Rev. Mod. Phys Vedral V et al 1997 Phys. Rev. Lett [16] Araki H ad Lieb E H 1970 Comm. Math. Phys [17] Vedral V ad Pleio M B 1998 Phys. Rev. A Wag C Z ad Fag M F 00 Chi. Phys [18] Rais E 1999 Phys. Rev. A [19] Wu S ad Zhag Y 000 Preprit quat-ph/ Liu J ad Fag M F 00 Chi. Phys