Generation of quantum entanglement between three level atoms via n coupled cavities
aa r X i v : . [ qu a n t - ph ] M a r Generation of quantum entanglement betweenthree level atoms via n coupled cavities R. Sufiani a,b ∗ and A. Darkhosh c † , a Department of Theoretical Physics and Astrophysics, University of Tabriz, Tabriz 51664, Iran. b Institute for Studies in Theoretical Physics and Mathematics, Tehran 19395-1795, Iran. c Physics Department, Faculty of Science, Atat¨urk University, Erzurum, Turkey.
September 3, 2018 ∗ E-mail:sofi[email protected] † E-mail:[email protected] oupled cavities Abstract
Based on two-photon exchange interaction between n coupled optical cavities each ofthem containing a single three level atom, the n -qubit and n -photonic state transfer isinvestigated. In fact, following the approach of Ref.[1], we consider n coupled cavitiesinstead of two cavities and generalize the discussions about quantum state transfer, pho-ton transition between cavities and entanglement generations between n atoms. Moreclearly, by employing the consistency of number of photons (the symmetry of Hamil-tonian), the hamiltonian of the system is reduced from 3 n dimensional space into 2 n dimensional one. Moreover, by introducing suitable basis for the atom-cavity state spacebased on Fourier transform, the reduced Hamiltonian is block-diagonalized, with 2 di-mensional blocks. Then, the initial state of the system is evolved under the correspondingHamiltonian and the suitable times T at which the initially unentangled atoms, becomemaximally entangled, are determined in terms of the hopping strength ξ between cavities. Keywords: coupled cavities, two-photon exchange, hopping strength, threelevel atoms, generation of entanglement, excitation and photon transfer,Fourier transformPACs Index: 03.65.Ud oupled cavities The quantum communication between several parts of a physical unit, is a crucial ingredientfor many quantum information processing protocols [2]. Schemes for the transfer of quantuminformation and the generation and distribution of entanglement have been designed and im-plemented, in the past years, in a number of physical systems (see for example [3]-[12]). Atomsand ions are particularly considered as tools for storing quantum information in their internalstates. Naturally, photons represent the best qubit carrier for fast and reliable communicationover long distances [13, 14]. Recently, using photons in order to achieve efficient quantum trans-mission between spatially distant atoms has considered in several works [1, 15, 16, 17, 18, 19].The basic idea, is to utilize strong coupling between optical cavities and the atoms. On theother hand, due to the ability of quantum entanglement as a resource for several quantuminformation processing tasks such as quantum communication, and certain quantum crypto-graphic protocols, the creation of quantum entanglement naturally arises as goals in nowadaysquantum control experiments in studying the nonclassical phenomena in quantum physics.One of the known models in quantum optics describing the atom-field interaction is theJaynes-Cummings Hamiltonian [20, 21]. In the study of three-level atoms, M. Alexanian and S.Bose [17] introduced a unitary transformation, whereby the three-level atom was reduced to acorresponding two-level atom of the Jaynes-Cummings type with two-photon instead of single-photon transitions. In Refs. [1, 18], entanglement properties of two and three atom-cavitysystems in which the cavities are coupled via two-photon exchange interaction, was analyzed indetail. Such results could set the pathway towards massively correlated multiphoton nonlinearquantum optical systems [22, 23], which are rapidly developing modern subjects nowadays.The motivation of interest to such systems is their promise in quantum switching, quantumcommunication and computation and quantum phase transition applications.In this paper, following the approach of Refs. [1, 17, 18], and introducing some suitable basis oupled cavities n coupled atom-cavity subsystems. More clearly, we consider a system of n spatially separatedoptical cavities, each containing a single three level atom, which are coupled to each otherwith two-photon exchange interaction. Our objective is to examine state transfer (atomicstate exchange or photon transition) within photon and atom subsystems and to considerpossible generation of the particle entanglement between the subsystems.The organization of the paper is as follows. In section 2, the model describing a systemof n identical atom-cavity subsystems is introduced. The main results of the paper suchas block-diagonalization of the Hamiltonian of the system, solving the Shr¨odinger equationfor time dependent probability amplitudes of the state of the system, and discussions aboutstate transfer (atomic excitation or photon transitions) and entanglement generation betweenatoms or photons, are given in this section. Sections 3 and 4 are respectively concerned withthe special cases of two and three identical coupled cavities. Paper is ended with a briefconclusion. n coupled cavities via two-photon ex-change interaction We will consider n identical cavities each containing one three-level atom, where the cavitiesare coupled via two photon hopping between them. In fact, we consider that the cavities arelocated at the nodes of the complete graph K n with n nodes and each cavity interacts with allof the other cavities via two-photon exchange.Let us first introduce the two-photon Hamiltonian obtained via an exact unitary transfor-mation introduced in Ref. [17]: H ( i ) = ¯ hωN ( i ) + E ( i )0 + ¯ hµσ ( i ) ee + ¯ hησ ( i ) gg + ¯ hλ ( σ ( i ) eg a i + σ ( i ) ge a † i ) oupled cavities N ( i ) = a † i a i + σ ( i ) ee − σ ( i ) gg + 1is a constant of motion for the i -th atom-cavity subsystem, i.e, we have [ H ( i ) , ˆ N ( i ) ] = 0 foreach i = 1 , , . . . , n . The operators a i and a † i are photon operators of the i -th cavity, and σ ( i ) ab = | a i ( i )( i ) h b | , for i = 1 , , . . . , n denote the atomic transition operators for the i -th cavityreferring to either the ground (g) or excited (e) state. Now, the Hamiltonian for the n cavitiesis given by: H = n X i =1 ( H ( i ) − H ( i )0 ) + ¯ hξ n X i,j =1; i 1) + ( E g + E e ) / , with E g , E e being the energies of the ground and exited states, respectively. The last termin the Hamiltonian (2-1) is the two-photon exchange interaction between the cavities, char-acterized by the hopping rate ξ . The parameters E , µ, η and λ are the free energies of thesubsystems written in the notation of Ref. [17] and we do not need their clear definitions inthe present paper. All of these parameters depend on the photon number in the correspondingcavities and so, on the cavity-mode intensity through the eigenvalues of the operator ˆ N ( i ) .The operator ˆ N = P ni =1 ˆ N ( i ) commutes with the Hamiltonian (2-1) and so we can reducethe Hamiltonian to the subspace spanned with the eigenstates of ˆ N and consider the timeevolution of the states in this subspace. For a given eigenspace of ˆ N with eigenvalue N , themaximum possible number of photons in a cavity is N when the corresponding atom is in theground state, which occurs when there are no photons present in the other cavities and theatoms are also in the ground state. Then, the total number of photons in the system will be N .The constant number of total photons determines the subspace or the manifold in which thestates evolve in time (the initial state of the system determines the constant number N ). Wewill consider the manifold with N = 2. In this case, each single atom-cavity system can take oupled cavities | g, i , | g, i or | e, i , and so, the total possible states that thesystem of n -cavities can take, are 3 n states. Due to the consistency of total N = 2, the onlypossible states which we can have, are 2 n states instead of 3 n ones. In fact, these 2 n statesare eigenstates of ˆ N with eigenvalue 2, and the 3 n -dimensional Hamiltonian H is reducedto 2 n -dimensional one in the bases which span the eigenspace of ˆ N with the correspondingeigenvalue 2. The bases states that span this subspace or manifold, are given by: | c i i = | g, i . . . | g, i | g, i | {z } i − th | g, i . . . | g, i , | a i i = | g, i . . . | g, i | e, i | {z } i − th | g, i . . . | g, i , for i = 0 , , . . . , n − 1. Indeed, these bases span the eigenspace of ˆ N with eigenvalue 2, i.e., wehave ˆ N ( α | c i i + β | a i i ) = 2( α | c i i + β | a i i ). Therefore, the general time dependent state of the n -cavity system is given by | ψ ( t ) i = n − X i =0 ( C i ( t ) | c i i + A i ( t ) | a i i ) . (2-2)Then, one can easily show that, by considering the order of bases as | c i , | a i , . . . , | c n − i , | a n − i ,the Hamiltonian H takes the following direct product form H = I n ⊗ θ tan θ tan θ + 2 ξ ( J n − I n ) ⊗ , (2-3)where, I n is n × n identity matrix and J n is the all one matrix of order n . The quantitytan θ is given by tan θ = √ r with r = g g , where g and g are the atom-photon couplingconstants in the three-level atom. In writing the above equation, the dimensionless time[( E +0 − E − ) cos θ ] t/ ¯ h → t and dimensionless hopping constant ¯ hξ/ [( E +0 − E − ) cos θ ] → ξ have introduced (see Ref.[1, 18] for the cases n = 2 and n = 3 coupled cavities), where E +0 and E − are eigenvalues associated with eigenvectors | ψ +0 i ( i ) = sin θ | e, i + cos θ | g, i and | ψ − i ( i ) = cos θ | e, i + sin θ | g, i of H ( i ) , respectively. oupled cavities J n has eigenvalues 0, and n (due to the fact that J n = nJ n ), and is diagonalized by discrete Fourier transform F defined as F kl := √ n ω kl for k, l =0 , , . . . , n − 1, where ω = exp ( πin ) is the n -th root of unity. Therefore, by introducing thenew Fourier transformed bases {| c i i ′ , | a i i ′ } n − l =0 as: | c l i ′ := 1 √ n n − X i =0 ω li | c i i , | a l i ′ := 1 √ n n − X i =0 ω li | a i i (2-4)and considering the ordering {| c i ′ , | a i ′ ; . . . ; | c n − i ′ , | a n − i ′ } , the Hamiltonian (2-3) takes thefollowing block diagonalized form: H = I n ⊗ θ tan θ tan θ + 2 ξ diag ( n − , − , . . . , − ⊗ , (2-5)where, diag ( n − , − , . . . , − 1) is the n × n diagonal matrix with diagonal entries as n − − i ¯ h ∂∂t | ψ i = H | ψ i , theequations of motion are given by: i ˙ C ′ = [1 + 2 ξ ( n − C ′ + tan θ A ′ ; i ˙ A ′ = tan θ C ′ + tan θ A ′ , and i ˙ C ′ l = (1 − ξ ) C ′ l + tan θ A ′ l ; i ˙ A ′ l = tan θ C ′ l + tan θ A ′ l , (2-6)for l = 1 , , . . . , n − 1. The equations (2-6) can be exactly solved for any value of tan θ .Here we take the ratio of atomic couplings in the three level atoms as r = 1 / √ θ = 1. Substituting tan θ = 1 in (2-6) and solving the corresponding differentialequations, one can obtain C ′ ( t ) = e − i [1+ ξ ( n − t q ξ ( n − { [ q ξ ( n − cos t q ξ ( n − − iξ ( n − 1) sin t q ξ ( n − ] C ′ (0) − oupled cavities i sin t q ξ ( n − A ′ (0) } ,A ′ ( t ) = e − i [1+ ξ ( n − t q ξ ( n − { [ q ξ ( n − cos t q ξ ( n − + iξ ( n − 1) sin t q ξ ( n − ] A ′ (0) − i sin t q ξ ( n − C ′ (0) } (2-7)where, for l = 1 , , . . . , n − C ′ l ( t ) = e − i (1 − ξ ) t √ ξ { [ q ξ cos t q ξ + iξ sin t q ξ ] C ′ l (0) − i sin t q ξ A ′ l (0) } ,A ′ l ( t ) = e − i (1 − ξ ) t √ ξ { [ q ξ cos t q ξ − iξ sin t q ξ ] A ′ l (0) − i sin t q ξ C ′ l (0) } . (2-8)By using (2-4), one can obtain the time dependence of the coefficients C i ( t ) and A i ( t ) of thestate of the system in (2-2) via the inverse Fourier transform as, C i ( t ) = 1 √ n n − X l =0 ω − li C ′ l ( t ) ,A i ( t ) = 1 √ n n − X l =0 ω − li A ′ l ( t ) . (2-9)It should be pointed out that, one can evaluate the probabilities associated with thestate of the system as a superposition of atomic states | a i i , and that of photonic states | c i i , denoted by P a ( t ) and P c ( t ), respectively. For instance, considering the initial state | ψ (0) i = √ n ( | g, i| g, i ... | g, i + | g, i| g, i| g, i ... | g, i + . . . + | g, i ... | g, i| g, i ), with initialconditions A l (0) = 0 and C l (0) = √ n for all l = 0 , , ..., n − 1, with the aid of Eqs. (2-2) and(2-8), we obtain P c ( t ) = n − X l =0 | C l ( t ) | = n − X l =0 | C ′ l ( t ) | = 1 n [1 + ξ ( n − ] { ξ ( n − + cos t q ξ ( n − } + n − n (1 + ξ ) { ξ + cos t q ξ } ,P a ( t ) = n − X l =0 | A l ( t ) | = n − X l =0 | A ′ l ( t ) | = 1 − P c ( t ) . (2-10)where, in the second equality in P c ( t ) and that of P a ( t ), we have used the fact that the Fouriertransform is unitary and so dose not change the norm of vectors. The above result indicates oupled cavities ξ → ∞ , we have P c ( t ) ≃ t , i.e., for large enough ξ ,all of the atoms will be at their ground state | g i at every time t . One should notice that for large values of the hopping strength, i.e., ξ ≫ , the evaluatedcoefficients C ′ i ( t ) and A ′ i ( t ) in (2-7) and (2-8) take the form C ′ ( t ) ≃ e − i [1+ ξ ( n − t { e − iξ ( n − t C ′ (0) − i sin ξ ( n − tξ ( n − A ′ (0) } ,A ′ ( t ) ≃ e − i [1+ ξ ( n − t { e iξ ( n − t A ′ (0) − i sin ξ ( n − tξ ( n − C ′ (0) } ,C ′ l ( t ) ≃ e − i (1 − ξ ) t { e iξt C ′ l (0) − i sin ξtξ A ′ l (0) } ,A ′ l ( t ) ≃ e − i (1 − ξ ) t { e − iξt A ′ l (0) − i sin ξtξ C ′ l (0) } ; l = 1 , , . . . , n − . (2-11)Neglecting also the second terms in the above approximations, we get C ′ ( t ) ≈ e − iξ ( n − t C ′ (0) ,C ′ l ( t ) ≈ e iξt C ′ l (0) , l = 1 , . . . , n − ,A ′ l ( t ) ≈ A ′ l (0) , l = 0 , , . . . , n − , and so by using (2-9), we obtain C l ( t ) ≈ e iξt n { e − iξnt n − X k =0 C k (0) + n − X k =0 [ n − X i =1 ω ( k − l ) i ] C k (0) } = e iξt n n − X k =0 ( e − iξnt − nδ k,l ) C k (0) ,A l ( t ) ≈ A l (0); for l = 0 , , . . . , n − n -th root of unity ω , we have P n − i =0 ω ( k − l ) i = nδ k,l and so P n − i =1 ω ( k − l ) i = nδ k,l − 1. The above results, are in correspondencewith those of Refs. [1, 18] for the special cases n = 2 and n = 3. Moreover, the relations (2-12)indicate that in the limit of large hopping strength, the state associated with the initially oupled cavities A l (0) = 0, for l = 0 , , . . . , n − 1, remainseffectively unentangled forever.In the limit of small hopping ξ ≪ , the equations (2-7) and (2-8) lead to the followingcoefficients C ′ i ( t ) and A ′ i ( t ) C ′ ( t ) ≃ e − i [1+ ξ ( n − t { [cos t − iξ ( n − 1) sin t ] C ′ (0) − i sin tA ′ (0) } ,A ′ ( t ) ≃ e − i [1+ ξ ( n − t { [cos t + iξ ( n − 1) sin t ] A ′ (0) − i sin tC ′ (0) } ,C ′ l ( t ) ≃ e − i (1 − ξ ) t { [cos t + iξ ( n − 1) sin t ] C ′ l (0) − i sin tA ′ l (0) } ,A ′ l ( t ) ≃ e − i (1 − ξ ) t { [cos t − iξ ( n − 1) sin t ] A ′ l (0) − i sin tC ′ l (0) } ; l = 1 , , . . . , n − . (2-13)Now, by neglecting the terms proportional to ξ , the above approximations read as C ′ ( t ) ≈ e − i [1+ ξ ( n − t { cos tC ′ (0) − i sin tA ′ (0) } ,A ′ ( t ) ≈ e − i [1+ ξ ( n − t { cos tA ′ (0) − i sin tC ′ (0) } ,C ′ l ( t ) ≈ e − i (1 − ξ ) t { cos tC ′ l (0) − i sin tA ′ l (0) } ,A ′ l ( t ) ≈ e − i (1 − ξ ) t { cos tA ′ l (0) − i sin tC ′ l (0) } ; l = 1 , , . . . , n − . (2-14)Then, by using (2-9), one can obtain for l = 0 , , . . . , n − C l ( t ) ≈ e − i (1 − ξ ) t n n − X k =0 ( e − iξnt − nδ k,l )(cos tC k (0) − i sin tA k (0)) ,A l ( t ) ≈ e − i (1 − ξ ) t n n − X k =0 ( e − iξnt − nδ k,l )(cos tA k (0) − i sin tC k (0)) , (2-15)It could be noted that for times such that ξnt ≪ 1, also for times such that ξt = kπn with k ∈ Z , the above result leads to C l ( t ) ∼ = e − i (1 − ξ ) t (cos tC l (0) − i sin tA l (0)) and A l ( t ) ∼ = e − i (1 − ξ ) t (cos tA l (0) − i sin tC l (0)), so that we have | C l ( t ) | + | A l ( t ) | = | C l (0) | + | A l (0) | andso, there is no exchange between the cavities. On the other hand, for the times such that oupled cavities ξt = (2 l +1) πn , with l ∈ Z , the exchange between the cavities (excitation or photon trans-fer) can be achieved. For instance, in the case of two cavities n = 2, for the initial state | ψ (0) i = | a i = | e, i| g, i with initial conditions A (0) = 1 and C (0) = A (0) = C (0) = 0,by using (2-15), we obtain at times t ≃ (2 l +1) π ξ , C ( t ) = A ( t ) = 0, C ( t ) = − ie − i (1 − ξ ) t sin t and A ( t ) = e − i (1 − ξ ) t cos t , so that we have | ψ ( t ) i = e − i (1 − ξ ) t (cos t | g, i| e, i − i sin t | g, i| g, i ).The results of this section can be used in order to discuss about qubit state transfer, photontransition and entanglement generation between the atoms. In order to clarify that, how onecan discuss these arguments, we will consider the special cases of two and three identicalcavities in the next sections in details. n = 2 For two cavities ( n = 2), by using the relations (2-7)-(2-9), one can calculate C ( t ) = C ′ + C ′ √ e − it √ ξ { [ q ξ cos ξt cos t q ξ − ξ sin ξt sin t q ξ ] C (0) − i [ q ξ sin ξt cos t q ξ + ξ cos ξt sin t q ξ ] C (0) − i sin t q ξ (cos ξtA (0) − i sin ξtA (0)) } ,C ( t ) = C ′ − C ′ √ e − it √ ξ {− i [ q ξ sin ξt cos t q ξ + ξ cos ξt sin t q ξ ] C (0)+[ q ξ cos ξt cos t q ξ − ξ sin ξt sin t q ξ ] C (0) − i sin t q ξ ( − i sin ξtA (0)+cos ξtA (0)) } ,A ( t ) = A ′ + A ′ √ e − it √ ξ { [ q ξ cos ξt cos t q ξ + ξ sin ξt sin t q ξ ] A (0) − i [ q ξ sin ξt cos t q ξ − ξ cos ξt sin t q ξ ] A (0) − i sin t q ξ (cos ξtC (0) − i sin ξtC (0)) } ,A ( t ) = A ′ − A ′ √ e − it √ ξ {− i [ q ξ sin ξt cos t q ξ − ξ cos ξt sin t q ξ ] A (0)+ oupled cavities q ξ cos ξt cos t q ξ + ξ sin ξt sin t q ξ ] A (0) − i sin t q ξ ( − i sin ξtC (0)+cos ξtC (0)) } . (3-16)For instance, for the initial state | ψ i = √ ( | c i + | c i ) = √ ( | g, i| g, i + | g, i| g, i ), wehave the initial conditions C (0) = C (0) = √ and A (0) = A (0) = 0. Then, by using therelations (3-16), the evolved state of the system will take the form | ψ ( t ) i = e − i (1+ ξ ) t q ξ ) { [ q ξ cos ξt cos t q ξ − iξ sin t q ξ ]( | c i + | c i ) − i sin t q ξ ( | a i + | a i ) } . Now, in order to investigate generation of entanglement between two atoms, we can evaluatethe density matrix ρ a associated with the atoms as ρ a ( t ) = T r c ( | ψ ( t ) ih ψ ( t ) | ) = 12(1 + ξ ) { ξ + cos t q ξ )( | g i (1) h g | ⊗ | g i (2) h g | )+sin t q ξ ( | e i (1) h e | ⊗ | g i (2) h g | + | g i (1) h g | ⊗ | e i (2) h e | + | e i (1) h g | ⊗ | g i (2) h e | + | g i (1) h e | ⊗ | e i (2) h g | ) } where, T r c denotes the partial trace over the photonic states | , i , | , i and | , i . Now, fora given hopping parameter ξ , one can use the Peres-Horodecki criteria [24, 25] known alsoas positive partial transpose (PPT) criteria, in order to determine that for which times t , thestate ρ a ( t ) is entangled, and particularly we can obtain the time T at which the perfect transferof photonic entanglement to the atomic one, is achieved. To this end, we choose the order ofatomic basis as | g, g i , | g, e i , | e, g i and | e, e i , so that the partial transpose of the atomic statetakes the following matrix form( ρ a ( t )) T = ξ + cos t √ ξ ) 0 0 sin t √ ξ t √ ξ t √ ξ t √ ξ . The corresponding eigenvalues of ( ρ a ( t )) T are given by λ = sin t √ ξ with double degen-eracy, and λ ± = ( ξ + cos t √ ξ ) ± q ( ξ + cos t √ ξ ) + sin t √ ξ . Therefore, theeigenvalue λ − is clearly negative, except for times t = kπ √ ξ , k ∈ Z , where the atomic state oupled cavities ρ a is separable. In order to evaluate the amount of entanglement of the atomic state ρ a , onecan calculate the corresponding concurrence [26], as C ( ρ a ( t )) = sin t √ ξ ξ . Then, for times T = (2 l +1) π √ ξ , l ∈ Z , (or in the suitable units, T = (2 l +1) π √ ξ ′ with ξ ′ = hξE +0 − E − ) the maximumvalue of the atomic entanglement is achieved and the corresponding concurrence takes its max-imum value C max = ξ , where the atomic density matrix will be maximally entangled forsmall hopping ξ → ξ ). Moreover, this result indicates that,for large hopping strength ξ → ∞ , we have C ( ρ a ( t )) → | ψ (0) i = | e, i| g, i with C (0) = C (0) = A (0) = 0 and A (0) = 1. The equations (3-16) give C ( t ) = − ie − it √ ξ sin t q ξ cos ξt,C ( t ) = − e − it √ ξ sin t q ξ sin ξt,A ( t ) = e − it √ ξ { q ξ cos t q ξ cos ξt + ξ sin t q ξ sin ξt } ,A ( t ) = − ie − it √ ξ { q ξ cos t q ξ sin ξt − ξ sin t q ξ cos ξt } , so that, one obtains | C ( t ) | + | C ( t ) | = sin t √ ξ ξ , | A ( t ) | + | A ( t ) | = ξ + cos t √ ξ ξ . Therefore, in the limit of large hopping ξ , we have | A ( t ) | + | A ( t ) | → A ( t ) and A ( t ).In addition, one can discuss two photon transfer from the first cavity to the second one.To do so, we consider the initial state | ψ (0) i = | g, i| g, i with initial conditions C (0) = 1, C (0) = A (0) = A (0) = 0. Then, the equations (3-16) give C ( t ) = e − it √ ξ { q ξ cos ξt cos t q ξ − ξ sin ξt sin t q ξ } , oupled cavities C ( t ) = − ie − it √ ξ { q ξ sin ξt cos t q ξ + ξ cos ξt sin t q ξ } ,A ( t ) = − ie − it √ ξ cos ξt sin t q ξ ,A ( t ) = − e − it √ ξ sin ξt sin t q ξ . Now, for large enough ξ ≫ , we obtain C ( t ) = e − it cos 2 ξt, C ( t ) = − ie − it sin 2 ξt, A ( t ) = − ie − it sin 2 ξt ξ ∼ = 0 , A ( t ) = − e − it sin ξtξ ∼ = 0 . Then, after times T = (2 k +1) π ξ with k ∈ Z , we have | ψ ( T ) i = ( − k +1 ie − it | g, i| g, i with | C ( T ) | = 1 and so, two photons of the first cavity are transmitted to the other cavity,perfectly. The case n = 3 identical cavities can be considered similar to the case of n = 2, by using therelations (2-7)-(2-9). Here we consider only the limits of large and small hopping ξ . ξ ≫ In the limit of large ξ , we use the equation (2-12) to obtain C ( t ) ≈ { [ C (0) + C (0) + C (0)] e − iξt + [2 C (0) − C (0) − C (0)] e iξt } ,C ( t ) ≈ { [ C (0) + C (0) + C (0)] e − iξt + [ − C (0) + 2 C (0) − C (0)] e iξt } ,C ( t ) ≈ { [ C (0) + C (0) + C (0)] e − iξt + [ − C (0) − C (0) + 2 C (0)] e iξt } ; A l ( t ) ≈ A l (0) , for l = 0 , , . The above results are in correspondence with those of Ref.[18]. By considering the initial state | ψ (0) i = | c i = | g, i| g, i| g, i with initial conditions C (0) = 1, C (0) = C (0) = A (0) = oupled cavities A (0) = A (0) = 0, we obtain the evolved state after time t as | ψ ( t ) i ≈ { ( e − iξt + 2 e iξt ) | c i + ( e − iξt − e iξt )( | c i + | c i ) } . Then, the probability of observing two photons at the first cavity ( | C ( t ) | ) and that of ob-serving two photons at the two other cavities ( | C ( t ) | = | C ( t ) | ), are given respectively by | C ( t ) | = 19 [1 + 4(1 + cos 6 ξt )] , | C ( t ) | = | C ( t ) | = 29 [1 − cos 6 ξt ] , which indicates that for times T = (2 k +1) π ξ , with k ∈ Z , the corresponding two photons initiallylocated at the first cavity, are transmitted to one of the other cavities with equal probability . ξ ≪ In the limit of small hopping ξ ≪ , the equation (2-15) leads to the following results for threeidentical cavities: C ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tC (0) − i sin tA (0)]+( e − iξt − t ( C (0)+ C (0)) − i sin t ( A (0)+ A (0))] } ,C ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tC (0) − i sin tA (0)]+( e − iξt − t ( C (0)+ C (0)) − i sin t ( A (0)+ A (0))] } ,C ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tC (0) − i sin tA (0)]+( e − iξt − t ( C (0)+ C (0)) − i sin t ( A (0)+ A (0))] } ,A ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tA (0) − i sin tC (0)]+( e − iξt − t ( A (0)+ A (0)) − i sin t ( C (0)+ C (0))] } ,A ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tA (0) − i sin tC (0)]+( e − iξt − t ( A (0)+ A (0)) − i sin t ( C (0)+ C (0))] } ,A ( t ) ≈ e − i (1 − ξ ) t { ( e − iξt +2)[cos tA (0) − i sin tC (0)]+( e − iξt − t ( A (0)+ A (0)) − i sin t ( C (0)+ C (0))] } . Now, by considering for example the initial state | ψ (0) i = | a i = | e, i| g, i| g, i , withinitial conditions A (0) = 1 and C (0) = C (0) = C (0) = A (0) = A (0) = 0, we obtain C ( t ) ≈ − i sin te − i (1 − ξ ) t e − i (1 − ξ ) t e − iξt +2) , C ( t ) = C ( t ) ≈ − i sin te − i (1 − ξ ) t e − i (1 − ξ ) t e − iξt − , oupled cavities A ( t ) ≈ cos te − i (1 − ξ ) t e − iξt + 2) , A ( t ) = A ( t ) ≈ cos te − i (1 − ξ ) t e − iξt − . Therefore, after times t = kπ , with k ∈ Z , the probability amplitudes C ( t ), C ( t ) and C ( t )will be zero and we will have A ( kπ ) ≈ ( − k e − i (1 − ξ ) kπ ( e − iξkπ + 2) and A ( kπ ) = A ( kπ ) ≈ ( − k e − i (1 − ξ ) kπ ( e − iξkπ − ξ = l +13 k with large k ∈ Z and small l ∈ Z , the excitation of the first atom located at the first cavity,can be transmitted with equal probability , to one of the other two atoms. In summery, the quantum entanglement properties of n coupled atom-cavity systems via two-photon exchange interaction, was analyzed. 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Lett, 80, pp. 2245-2248, (1998). oupled cavities Figure CaptionFigure.1: Shows the concurrence C ( ρ ) of the atomic state ρ a ( t ) with respect to time, fordifferent values of hopping strength (a) ξ = 0 . 1, (b) ξ = 0 . 5, (c) ξ = 0 . ξξ