Entanglement dynamics for the double Tavis-Cummings model
aa r X i v : . [ qu a n t - ph ] F e b Entanglement dynamics for the double Tavis-Cummingsmodel
Zhong-Xiao Man , Yun-Jie Xia , and Nguyen Ba An , College of Physics and Engineering, Qufu Normal University, Qufu 273165, China Institute of Physics and Electronics, 10 Dao Tan, Thu Le, Ba Dinh, Hanoi, Vietnam School of Computational Sciences, Korea Institute for Advanced Study, 207-43 Cheon-gryangni 2-dong, Dongdaemun-gu, Seoul 130-722, Korea
Abstract.
A double Tavis-Cummings model (DTCM) is developed to simu-late the entanglement dynamics of realistic quantum information processingwhere two entangled atom-pairs AB and CD are distributed in such a waythat atoms AC are embedded in a cavity a while BD are located in anotherremote cavity b . The evolutions of different types of initially shared entan-glement of atoms are studied under various initial states of cavity fields. Theresults obtained in the DTCM are compared with that obtained in the dou-ble Jaynes-Cummings model (DJCM) [J. Phys. B , S45 (2007)] and aninteraction strength theory is proposed to explain the parameter domain inwhich the so-called entanglement sudden death occurs for both the DTCMand DJCM. PACS.
QICS.
Entanglement is not only a key concept to distinguish between the quantum and the classicalworlds, but has also been viewed as an indispensable resource to perform various intriguingglobal tasks in quantum computing and quantum information processing [1]. However, anotable characteristic of entanglement is its fragility in practical applications due to unavoid-able interaction with the environment. It is therefore of increasing importance to understandentanglement from its dynamical behaviors in realistic systems. As a rule for a global task,entanglement should be shared between different remote parties who participate in the task.There are cases like teleportation [2], remote state preparation [3], etc., in which each particleof a multipartite entangled state is distributed to a separate location. There are also casesin which the entangled particles should be distributed so that each location contains severalparticles. For example, in quantum secret communication protocol between Alice and Bob[4], an ordered N Einstein-Podolky-Rosen (EPR) pairs are to be shared in such a way thatAlice and Bob each holds one half of the pairs. That is, at Alice’s location there are N particles which interact with one environment while the other N partner-particles at Bob’slocation collectively interact with another environment. This scenario results in two inde-pendent local environments but each of them is common for one half of the N EPR pairs. Anatural question arises as to how such kind of particle-environment interactions degrade theoriginally prepared global entanglement. This question is of fundamental interest becauseany quantum protocol depends essentially on the quality of the shared entanglement. Asa first step to the problem, in this paper, we consider the case of N = 2 with two pairs ofentangled two-level atoms AB and CD prepared in one of the two types of Bell-like states,namely, | ψ (0) i IJ = cos( α ) | i IJ + sin( α ) | i IJ , (1)and | ϕ (0) i IJ = cos( α ) | i IJ + sin( α ) | i IJ , (2)where IJ ∈ { AB, CD } and | i ( | i ) is the atomic ground (excited) state.For the simplest case of N = 1 , i.e., either state (1) or state (2) is concerned for theinitial state of a single atom-pair, the so-called double Jaynes-Cummings model (DJCM)[5-12] has been extensively adopted to study this problem because it yields exact analyticalresults. In the DJCM, each of two entangled atoms is embedded in an independent cavityand locally interacts with it. The results obtained within the DJCM for the initial emptycavities are that for any value of α state (1) loses its entanglement only at discrete timemoments t l = ( l + 1 / π/g with l = 0 , , , ... and g the atom-cavity coupling constant,but for a certain domain of α state (2) may become separable at times smaller than t l and remains unentangled for some duration of time [6]. The latter phenomenon is referredto in the current literatures as entanglement sudden death (ESD) [13], which has beenexperimentally observed in [14,15]. An entangled state with ESD in evolution is less robust AC a BDb
FIG. 1: Schematic representation of two entangled atom-pairs AB and CD of which atoms A and C are located in cavity a but atoms B and D in another cavity b . than states without it, since ESD puts a limitation on the application time of entanglement.Therefore, studying ESD, especially conditions and parameter domains for its occurrence,is important from both theoretical and practical points of view. In Ref. [10] the DJCMis considered again and it is found that if the cavity fields are initially in Fock states withnonzero photon numbers then both atomic states | ψ (0) i and | ϕ (0) i would suffer from ESDfor all values of α. The DJCM was also investigated from other perspectives and it wasshown that the entanglement evolution of atoms is closely related to their energy variation[9] and there is a natural entanglement invariant demonstrating the entanglement transferamong all the system’s degrees of freedom [7].For the case of N = 2 involving two pairs of entangled atoms, the situation wouldbecome more complex than that of N = 1, because in each local environment there are twoatoms simultaneously interacting with it. When there are many atoms interacting resonantlywith a single-mode quantized radiation field of one and the same cavity the exact solutioncan be obtained by means of the so-called Tavis-Cummings model (TCM) [16]. Such asingle TCM was used in Refs. [17] and [18] to study entanglement dynamics of two atomsthat are initially prepared in a separable and entangled state, respectively. In this workwe develop the so-called double Tavis-Cummings model (DTCM) including four two-levelatoms A, B, C, D and two separate single-mode cavities a, b (see FIG. 1), which suffices forour purpose to study the entanglement dynamics for case of N = 2. In the DTCM, atoms A ( C ) and B ( D ) are initially prepared either in state (1) or (2), but atoms A and C ( B and D ) are located in cavity a ( b ) and interact with the cavity through the Tavis-CummingsHamiltonian. We study the entanglement dynamics of atom-pairs AB, CD, AC and BD bymeans of concurrence in dependence on the initial entanglement type of the atoms and onthe initial state of cavity fields. We compare our results obtained in the DTCM with thoseobtained in the DJCM and present an interaction strength theory to explain the parameterdomain in which the atom-pair exhibit ESD for both the DTCM and the DJCM.Our paper is organized as follows. In Sec. 2 we describe the DTCM and derive the exactanalytical expression for the reduced density matrix of the atomic subsystem. Section 3presents detailed analysis of atomic entanglement dynamics when the initial atom-pairs areprepared either in state (1) or state (2) and the initial cavity fields are prepared either inthe vacuum state, Fock state with a non-zero photon number or the thermal state. Finally,we conclude in Sec. 4. The total Hamiltonian of the system of four atoms
A, B, C, D and two cavities a, b (see FIG.1) in the DTCM can be written as a sum of two isolated Tavis-Cummings Hamiltonians H = H ACa + H BDb , (3)with H ACa = ω σ zA + σ zC ) + ωa + a + g X i = A,C ( aσ + i + a + σ − i ) , (4)and H BDb = ω σ zB + σ zD ) + ωb + b + g X i = B,D ( bσ + i + b + σ − i ) , (5)where ω ( ω ) is the frequency of the atom (cavity field mode), a ( a + ) is the annihilation(creation) operator of the field in cavity a, b ( b + ) is the annihilation (creation) operator ofthe field in cavity b, σ + i = | i ii h | ( σ − i = | i ii h | ) is the rising (lowering) operator for thetransition of atom i and g is the atom-cavity field coupling constant. Here, we are interestedin the resonant case with ω = ω [16]. The initial cavity fields are assumed to be either inthe vacuum state, the Fock state with a non-zero photon number or the thermal state. Thegeneral thermal field with its mean photon number n is a weighted mixture of Fock stateswhose density operator ρ F can be represented as ρ F = ∞ X n =0 P n | n i h n | , (6)with | n i the Fock state of n photons and P n is given by P n = n n (1 + n ) n +1 . (7)By virtue of the general thermal field defined above, through setting P n = δ nl in Eq. (6),we can also study the vacuum state ( l = 0) as well as any Fock states ( l >
0) of the fields.As for the initial states of atom-pairs AB and CD , we assume both of them to be eitherin state (1) or state (2). At t = 0 the total state involving the four atoms and two cavitiesreads ρ (0) = X i,j,k,l =0 ∞ X m,n =0 α i α j α k α l P am P bn | ik, m i ACaACa h jl, m | ⊗ | i ′ k ′ , n i BDbBDb h j ′ l ′ , n | , (8)where α ≡ sin α, α ≡ cos α and i ′ ( j ′ , k ′ , l ′ ) ≡ i ( j, k, l ) ⊕ ⊕ an additionmod 2) for state (1), while i ′ ( j ′ , k ′ , l ′ ) ≡ i ( j, k, l ) for state (2). The evolution operator U ACa ( BDb ) ( t ) = exp( − iH ACa ( BDb ) t ) for the local interaction of atoms AC ( BD ) with cav-ity a ( b ) was derived exactly in Ref. [17]. At any time t > ρ (0) evolves into ρ ( t ) = U ACa ( t ) U BDb ( t ) ρ (0) U + ACa ( t ) U + BDb ( t ) which can be represented as ρ ( t ) = X i,j,k,l =0 ∞ X m,n =0 α i α j α k α l P am P bn U ACa ( t ) | ik, m i ACaACa h jl, m | U + ACa ( t ) ⊗ U BDb ( t ) | i ′ k ′ , n i BDbBDb h j ′ l ′ , n | U + BDb . (9)Using the analytical expression of U ACa ( BDb ) ( t ) in [17] we have for U ACa | ik, m i ACa (similarlyfor U BDb | i ′ k ′ , n i BDb ) : U ACa ( t ) | ik, m i ACa = X p,q =0 X ik,pq ( m, τ ) | i ⊕ p, k ⊕ q i AC (cid:12)(cid:12)(cid:12) m − ( − i p − ( − k q E a (10)where the functions X ik,pq ( m, τ ) with τ = gt are given in Appendix A for various possible i, k, p, q. These functions satisfy the normalization condition X p,q =0 | X ik,pq ( m, τ ) | = 1 (11)for any i, k, m and τ. The reduced density matrix ρ ABCD ( t ) of the atomic subsystem can be obtained by tracingout ρ ( t ) over the cavity fields, i.e. ρ ABCD ( t ) = Tr ab ρ ( t ) = X i,j,k,l =0 α i α j α k α l E aAC ( | ik i ACAC h jl | ) ⊗ E bBD ( | i ′ k ′ i BDBD h j ′ l ′ | ) (12)where E cXY ( | ik i XY XY h jl | ) , with XY c = ACa or BDb, represents the map E cXY ( | ik i XY XY h jl | ) ≡ ∞ X m,m ′ =0 P cm h m ′ | U XY c ( t ) | ik, m i XY cXY c h jl, m | U + XY c ( t ) | m ′ i = ∞ X m =0 1 X r,s,u,v =0 P cm δ ( − i r − ( − k s, ( − j u − ( − l v × X ik,rs ( m, τ ) X ∗ jl,uv ( m, τ ) | i ⊕ r, k ⊕ s i XY XY h j ⊕ u, l ⊕ v | . (13)The explicit expressions of E cXY ( | ik i XY XY h jl | ) are given in Appendix B for various possible i, k, j, l. With the formulae derived in the previous section we are now in the position to analyzethe entanglement dynamics of any atom-pair. By using Eq. (12) we can readily get thereduced density matrix of any pair of atoms by tracing out ρ ABCD ( t ) over the degrees offreedom of the remaining atoms. In two-qubit domains, there exist a number of goodmeasures of entanglement such as concurrence [19] and negativity [20]. Although the variousentanglement measures may be somewhat different quantitatively [6], they are qualitativelyequivalent to each other in the sense that all of them are equal to zero for unentangledstates. Here we adopt Wootters’ concurrence [19] because of its convenience in definition,normalization and calculation. The concurrence C for any (reduced) density matrix ρ oftwo qubits is defined as C ( ρ ) = max { , q λ − q λ − q λ − q λ } , (14)where λ i ( λ ≥ λ ≥ λ ≥ λ ) are the eigenvalues of the matrix ζ = ρ ( σ y ⊗ σ y ) ρ ∗ ( σ y ⊗ σ y ) , with σ y a Pauli matrix and ρ ∗ the complex conjugation of ρ in the standard basis. Forseparate states C ( ρ ) = 0 , whereas for maximally entangled states C ( ρ ) = 1 . In particular,if ρ is of the X-form [21], ρ IJ = ̺ IJ ̺ IJ ̺ IJ ̺ IJ ̺ IJ ̺ IJ ̺ IJ ̺ IJ , (15)where ̺ IJkk are real positive and ̺ IJkl = (cid:16) ̺ IJlk (cid:17) ∗ are generally complex, then the concurrence(14) simplifies to C IJ = 2 max { , | ̺ IJ | − q ̺ IJ ̺ IJ , | ̺ IJ | − q ̺ IJ ̺ IJ } . (16)Since both states (1) and (2) of the atoms take on and preserve the X-form in their evolution,Eq. (16) is very useful throughout this work. | ψ (0) i type initial state for atom-pairs AB and CD We first consider the case when both the atom-pairs AB and CD are initially prepared instate (1). In accordance with Eq. (12) the reduced density matrix of the atomic subsystemat any time t is ρ ABCDI ( t ) = X i,j,k,l =0 α i α j α k α l E aAC ( | i, k i ACAC h j, l | ) ⊗ E bBD ( | i ⊕ , k ⊕ i BDBD h j ⊕ , l ⊕ | ) , (17)which can be evaluated straightforwardly via the map (13). Then the reduced density matri-ces of interest are ρ ABI ( t ) =Tr CD ρ ABCDI ( t ) , ρ CDI ( t ) =Tr AB ρ ABCDI ( t ) , ρ ACI ( t ) =Tr BD ρ ABCDI ( t )and ρ BDI ( t ) =Tr AC ρ ABCDI ( t ) . All of ρ ABI ( t ) , ρ CDI ( t ) , ρ ACI ( t ) and ρ BDI ( t ) have the X -form sothe corresponding concurrences are determined by Eq. (16). In the following we study thetime dependence of these concurrences for the fields in cavities a and b being initially in thevacuum state, the Fock state with a non-zero photon number or the general thermal state,respectively.In FIG.2 we plot C ABI (the same for C BDI due to symmetry) as functions of rescaledtime gt and α for the initial cavity fields being in the vacuum state. From FIG. 2 it istransparent that C ABI vanishes after a finite time of evolution and remains zero for someperiod of time before increasing again. This dynamics holds in the whole range of α. Acomparison between the DTCM and the DJCM [6] for the same initial preparation of the Α AB FIG. 2: The concurrence C AB ≡ C ABI ( t ) as functions of rescaled time gt and α for initially bothcavity fields are in the vacuum state and both atom-pairs AB and CD are in the | ψ (0) i (1) typestate in the DTCM. (b) C AB C AB gt (a) FIG. 3: The concurrence C AB ≡ C ABI ( t ) as a function of gt for various values of α for the sameinitial preparation of cavities and atoms as in Fig. 2 in (a) the DTCM and (b) the DJCM. cavities and atoms is shown in FIG. 3. Within the first cycle of evolution, in the DJCM(see FIG. 3b) C ABI vanishes at the moment t = π/ (2 g ) and grows up again right after t , while in the DTCM (see FIG. 3a) C ABI = 0 at a time shorter than t and remains so forsome time before reviving. This indicates that for one and the same empty cavity fields, | ψ (0) i type initial state of atoms does not undergo ESD in the DJCM but it does in theDTCM. Therefore, the atomic entanglement dynamics is model-dependent apart from theentanglement type itself. The physical interpretation behind such a clear distinction in thedynamical behaviors between the two models can be thought of as follows. If the cavitiesare empty, atoms in the ground state | i remain unchanged and only atoms in the excitedstate | i can interact with the cavity fields. Denoting by N | i the number of atoms thatmay be populated in state | i , the system-environment interaction can be classified intotwo regimes, “strong” and “weak” interaction regimes, depending on relative magnitudes of P ≥ and P < , where P ≥ ( P < ) is the probability that N | i ≥ N c ( N | i < N c ) with N c the numberof cavities. In the DTCM considered here and the DJCM considered in [6,7] it is clear that N c = 2 . We define the following convention: the strong interaction regime corresponds to P ≥ > P < , while P ≥ ≤ P < implies the weak interaction regime. In the DJCM the totalsystem state of two atoms A, B and two cavities a, b at t = 0 reads | ψ (0) i AB | i ab = cos α | i Aa | i Bb + sin α | i Aa | i Bb , (18)whereas in the DTCM the total system state of four atoms A, B, C, D and two cavities a, b at t = 0 reads | ψ (0) i AB | ψ (0) i CD | i ab = cos α | i ACa | i BDb + cos α sin α | i ACa | i BDb + sin α cos α | i ACa | i BDb + sin α | i ACa | i BDb . (19)From Eq. (18) it follows that there is always only one atom (namely, either atom A in thefirst term or atom B in the second term) being in state | i regardless of the value of α. Thatis, P < = 1 < P ≥ = 0 , resulting in the weak interaction regime in the DJCM for the wholerange of α. However, what is followed from Eq. (19) is that for any value of α there arealways two atoms (namely, either atoms A and C in the first term or atoms A and D in thesecond term or atoms C and B in the third term or atoms B and D in the fourth term)being in state | i . That is, P ≥ = 1 > P < = 0 , resulting in the strong interaction regime inthe DTCM regardless of the value of α. Therefore, it can be said that, when the cavitiesare initially prepared in the vacuum state, | ψ (0) i type initial state of atoms exhibits ESDin the strong interaction regime (i.e., in the DTCM) but it does not in the weak interactionregime (i.e., in the DJCM), independent of the parameter α. The case when the initial cavity fields are in a Fock state with a certain nonzero photonnumber is illustrated in FIG. 4. In this case not only atoms in state | i but also atoms instate | i , i.e., all the present atoms, can interact with the cavity fields so that the interaction0 m=n=10m=n=1 C AB gt m=n=0 FIG. 4: The concurrence C AB ≡ C ABI ( t ) as a function of gt for α = π/ | mn i ab and atom-pairs AB and CD are in the | ψ (0) i (1) typestate in the DTCM. C AB gtm=n=0.1m=n=1 FIG. 5: The concurrence C AB ≡ C ABI ( t ) as a function of gt for α = π/ m , n and atom-pairs AB and CD are in the | ψ (0) i (1) type state in the DTCM. regime is always strong resulting in ESD for whatever values of α. A remarkable feature isthat C ABI decays quicker and reaches zero in a shorter time for a larger initial number ofphotons in the cavities. The underlying physics for that feature is the intensification of thesystem-environment effective interaction with the increase of photon number contained inthe cavities.1 Α BD FIG. 6: The concurrence C BD ≡ C BDI ( t ) as functions of gt and α for initially the cavity fields arein the Fock state | i ab and both atom-pairs AB and CD are in the state (1) in the DTCM. C B D gt FIG. 7: The concurrence C BD ≡ C BDI ( t ) as a function of gt for various values of α with the sameinitial preparation of cavity fields and atom-pairs as in Fig. 6 in the DTCM. Figure 5 plots the evolution of C ABI for the cavity fields being initially in the thermalstate. The entanglement dynamics looks chaotic due to the nature of the thermal fields. Ascan be seen from FIG. 5, the larger the mean photon number (corresponding to the highertemperature) the shorter the death time of C ABI and the longer its revival time.At this point let us study the dynamics of the two atoms that are located in one and thesame cavity. These are atoms A and C in cavity a and atoms B and D in cavity b. Suchatoms in the same cavity are absolutely uncorrelated at the beginning and also there are no2 Α BD FIG. 8: The concurrence C BD ≡ C BDI ( t ) as functions of gt and α for initially the cavity fields arein the thermal state with the mean photon numbers m = n = 1 and both atom-pairs AB and CD are in the state (1) in the DTCM. direct interactions between them during the entire course of evolution, in accordance with theproblem Hamiltonians (4) and (5). However, an effective (indirect) atom-atom interactionis induced for t > | i or | i ( | i ), then they always get entangled with each other (remain unentangled) regardlessof the nature of the cavity fields. But, if the atomic initial state is | i , then the field inthe vacuum state leaves the atoms unentangled and the field in a Fock state with a non-zero photon number or thermal state can entangle them. Here, in the DTCM, at variancewith the situation considered in Ref. [17], at t = 0 the atoms in a cavity, though beingindependent of each other, are entangled with other atoms in another cavity. That is, we haveat t = 0 in cavity a ( b ) a mixed state ρ ACI (0) =Tr BD ρ ABCDI (0) = P i,j =0 α i α j | i, j i ACAC h i, j | ( ρ BDI (0) =Tr AC ρ ABCDI (0) = P i,j =0 α i α j | i ⊕ , j ⊕ i BDBD h i ⊕ , j ⊕ | ) , instead of a purestate as in Ref. [17]. Figure 6 plots the concurrence C BDI as functions of gt and α withthe initial fields in both cavities containing just one photon. This figure shows that theentanglement dynamics of the atoms is sensitive to α, as it should be. For example, inthe region of α ∈ [0 , . π ] atoms B and D can get entangled, but for α around π/ α = 0 (i.e., ρ BDI (0) = | i BDBD h | ) and α = π/ ρ BDI (0) = | i BDBD h | ) are concerned. To get more insight into the effect of α onatomic entanglement generation we show in FIG. 7 a 2D plot of C BDI as a function of gt withthe initial cavity fields in the Fock states | , i ab for various values of α. When α = 0 (i.e., ρ BDI (0) = | i BDBD h | ) , the entanglement of B and D emerges immediately from t = 0 . Nevertheless, when α > α the longer the delaytime of entanglement generation. Such phenomena of delayed entanglement during the timeevolution can be called “entanglement sudden birth” (ESB) [22]. The effect of thermal fieldson inducing entanglement between atoms B and D is drawn in FIG. 8 with the cavity meanphoton numbers m = n = 1 , which agrees well with the result in Ref. [17] for α = 0 . Sincethe thermal state is a weighted mixture of Fock states (see Eq. (6)), it is a chaotic state withminimum information and so its effect is generally irregular. In comparison with the case of“corresponding” Fock states | , i ab one sees that the region of α allowing entanglement ofatoms is much shrunk and the amount of generated entanglement is very small. The plotsof C ACI can be obtained from those of C BDI by making a change α → α + π/ . | ϕ (0) i type initial state for atom-pairs AB and CD We next consider the case when both atom-pairs AB and CD are initially prepared instate (2). In accordance with Eq. (12) the reduced density matrix of the atomic subsystemat any time t is ρ ABCDII ( t ) = X i,j,k,l =0 α i α j α k α l E aAC ( | i, k i ACAC h j, l | ) ⊗ E bBD ( | i, k i BDBD h j, l | ) . (20)In FIG.9 we plot C ABII (the same for C BDII due to symmetry) versus gt and α for the initialempty cavity fields. It is visual from this figure that ESD occurs but not in the whole rangeof α, in clear contrast with the case shown in FIG. 2 when both the atom-pairs AB and CD are initially prepared in state (1). To derive the constraint on α that triggers ESD let uslook at the total system state at t = 0 : | ϕ (0) i AB | ϕ (0) i CD | i ab = cos α | i ACa | i BDb + cos α sin α | i ACa | i BDb + sin α cos α | i ACa | i BDb + sin α | i ACa | i BDb . (21)4Obviously, the probability that all the four atoms are in state | i is cos α, the probabilitythat only two atoms (namely, either atoms A and B or atoms C and D ) are in state | i is 2 cos α sin α and the probability that none of the atoms are in state | i (i.e., all theatoms are in state | i ) is sin α. That is, P ≥ = cos α + 2 cos α sin α and P < = sin α. Asmentioned in the previous subsection, the condition for the occurrence of ESD is that theinteraction regime is strong, i.e., P ≥ > P < . So, the values of α for which ESD occurs shouldsatisfy the constraint sin α < √ . (22)Noticeably, this constraint is not coincident with that one in the DJCM for which the initialtotal system state reads | ϕ (0) i AB | i ab = cos α | i Aa | i Bb + sin α | i Aa | i Bb . (23)As followed from Eq. (23), the probability that the two atoms are in state | i is cos α and the probability that none of the atoms are in state | i is sin α. That is, P ≥ = cos α,P < = sin α and thus the values of α, for which the system-environment interaction regimeis strong (i.e., ESD occurs) in the DJCM, satisfy the constraintsin α < . (24)The constraints (22) and (24) imply that the α -parameter domain in which the atoms sufferfrom ESD is wider in the DTCM than in the DJCM.The case for the initial cavity fields being in a Fock state | i ab is plotted in FIG. 10.A remarkable feature as compared with the vacuum fields case in FIG. 9 is that here ESDoccurs in the whole range of α. Again, the physical reason for this is that in the presence ofinitial photons all the atoms are in interaction with the cavity fields (i.e., not only atoms instate | i but also those in state | i interact with the cavity fields).In FIG. 11 we plot C ABII as a function of gt for the initial fields in a thermal state withdifferent mean photon numbers for a given value of α. Comparing FIG. 11 with FIG. 5signals that with relatively small mean photon numbers (e.g., m = n = 0 .
1) the signatureof ESD is less pronounced for the case when the initial atoms are prepared in state (2) thanin state (1).The entanglement generation dynamics of the atomic pairs AC and BD is similar to thecase considered in the preceding subsection and thus will not be iterated here.5 Α AB FIG. 9: The concurrence C AB ≡ C ABII ( t ) as functions of gt and α for initially both cavity fieldsare in the vacuum state and both atom-pairs AB and CD are in the | ϕ (0) i (2) type state in theDTCM. Α AB FIG. 10: The concurrence C AB ≡ C ABII ( t ) as functions of gt and α for initially the cavity fields arein the Fock state | i ab and both atom-pairs AB and CD are in the | ϕ (0) i (2) type state in theDTCM. In conclusion, we have, by means of concurrence, studied the entanglement dynamics ofthe DTCM motivated by certain realistic quantum information processing. The system iscomposed of four two-level atoms
A, B, C, D and two spatially separated single-mode cavities6 m=n=0.1 C AB gt m=n=1 FIG. 11: The concurrence C AB ≡ C ABII ( t ) as a function of gt for α = π/ m , n and both atom-pairs AB and CD are in the | ϕ (0) i (2) type state in the DTCM. a, b . Initially, atom-pairs AB and CD are prepared either in Bell-like state | ψ (0) i (1) or | ϕ (0) i (2), while both cavities are prepared either in the vacuum state, the Fock state withnon-zero photon numbers or the thermal sate. Independent atoms A, C ( B, D ) that belongto different entangled atom-pairs are embedded in one and the same cavity a ( b ) and interactwith it through the Tavis-Cummings Hamiltonian.For the vacuum fields the | ψ (0) i type initial state of atom-pairs AB and CD displaysESD for the whole value range of the parameter α which represents the initial entanglementdegree of AB and CD . This result is in sharp contrast with the DJCM for which ESDdoes not occur at all for whatever values of α [6,7]. As for the | ϕ (0) i type initial state ofatom-pairs AB and CD , ESD only occur for the value of α such that sin α < / √
2, whichis wider than that in the DJCM where ESD occurs just for α such that sin α < / α for which ESD occurs) in both the DTCMand DJCM can be explained via the interaction strength theory according to which ESDoccurs (does not occur) in the strong (weak) system-environment interaction regime. Theinteraction regime is identified by the number of atoms that can have interaction with thecavities, which is determined by the relative magnitudes of P ≥ and P < defined in subsection3.1. Remarkably, the interaction strength theory turns out to apply also for the so-calledtriple Jaynes-Cummings model [23] for GHZ-like atomic states as well as for the case of7multiple dissipative environments with multiqubit GHZ-like atomic states [24,25].We have shown that the non-vacuum environments of cavities have great effects on theappearance of ESD for atoms. That is, when the cavity fields are initially in the Fockstate with a non-zero photon number or the general thermal state, ESD always happens foratom-pairs AB and CD regardless of the entanglement type they are prepared. Moreover,the more photon number in the Fock state or the greater the mean photon number in thethermal state the quicker the entanglement decay rate, i.e., the sooner the time of ESDoccurrence. In terms of the interaction strength theory, these properties are explained bythe physical fact that in the presence of nonzero (mean) photon number the interactionregime is always strong because all the atoms (i.e., not only those in the excited state as inthe case of empty cavities) can interact with the fields. Thus, the actual system-environmentinteraction strength is now identified by the number of excitation which in these cases isproportional to the total number of both atoms and photons.We have also studied creation of entanglement between initially uncorrelated atoms A and C in cavity a ( B and D in cavity b ) . Compared to the case of α = 0 considered inRef. [17] here we showed that for α = 0 there appears the so-called entanglement suddenbirth, i.e., the formation of atomic entanglement does not take place at once as the systemevolves but emerges suddenly at some delayed time, which is dependent on the value of α. The DTCM presented in this work could be extended to the general multiple case where twogroups of multipartite entangled atoms are distributed in such a way that every two atomsfrom different group are located in the same environment. In this way, we can study not onlythe pairwise entanglement of atoms between any two nodes (cavities or local environments)via concurrence but also the entanglement of any atomic bipartition by means of negativity.These studies can reveal the degraded properties of various multipartite entangled state andthus be useful for the large-scale quantum information processing.Z.X.M. and Y.J.X. are supported by National Natural Science Foundation of China underGrant No. 10774088. N.B.A. acknowledges support from a NAFOSTED project No. NCCB-2009 and from the KIAS Scholar program.8
APPENDIX A: THE EXPLICIT EXPRESSIONS OF X ik,pq ( m, τ ) The functions X ik,pq ( m, τ ) appearing in Eq. (10) for all possible i, k, p, q are given by X , ( m, τ ) = m + 12 m + 3 [cos( q m + 3) τ ) −
1] + 1 , (A1) X , ( m, τ ) = X , ( m, τ ) = − i s m + 12(2 m + 3) sin( q m + 3) τ ) , (A2) X , ( m, τ ) = q ( m + 1)( m + 2)2 m + 3 [cos( q m + 3) τ ) − , (A3) X , ( m, τ ) = X , ( m, τ ) = − i s m m + 1) sin( q m + 1) τ ) , (A4) X , ( m, τ ) = X , ( m, τ ) = 12 [cos( q m + 1) τ ) + 1] , (A5) X , ( m, τ ) = X , ( m, τ ) = 12 [cos( q m + 1) τ ) − , (A6) X , ( m, τ ) = X , ( m, τ ) = − i s m + 12(2 m + 1) sin( q m + 1) τ ) , (A7) X , ( m, τ ) = q m ( m − m − q m − τ ) − , (A8) X , ( m, τ ) = X , ( m, τ ) = − i s m m −
1) sin( q m − τ ) (A9)and X , ( m, τ ) = m m − q m − τ ) −
1] + 1 . (A10) APPENDIX B: THE EXPLICIT EXPRESSIONS OF E cXY ( | ik i XY XY h jl | ) The expressions of the map E cXY ( | ik i XY XY h jl | ) , with XY c = ACa or BDb, appearingin Eq. (13) for all possible i, k, j, l are given by E cXY ( | i XY XY h | ) = ∞ X m =0 P cm h | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | i , (B1)9 E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm h X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY ( h | + h | ) | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | i , (B2) E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm h X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | i , (B3) E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | , (B4) E cXY ( | i XY XY h | ) = ∞ X m =0 P cm h | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | i , (B5) E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm h | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | i , (B6)0 E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm h X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | i , (B7) E cXY ( | i XY XY h | ) = ∞ X m =0 P cm h | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | | i XY XY h | i , (B8) E cXY ( | i XY XY h | ) = E cXY ( | i XY XY h | ) ∗ = ∞ X m =0 P cm h X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | + X , ( m, τ ) X ∗ , ( m, τ ) | i XY XY h | i (B9)and E cXY ( | i XY XY h | ) = ∞ X m =0 P cm h | X , ( m, τ ) | | i XY XY h | + | X , ( m, τ ) | ( | i + | i ) XY XY ( h | + h | )+ | X , ( m, τ ) | | i XY XY h | i . (B10) References
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