Entanglement for Two-Qubit Extended Werner-Like States: Effect of Non-Markovian Environments

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1 Commun. Theor. Phys. (Beijing, China) 5 (2010) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 3, September 15, 2010 Entanglement for Two-Qubit Extended Werner-Like States: Effect of Non-Markovian Environments SHAN Chuan-Jia ( ), LIU Ji-Bing ( ÍÏ), CHEN Tao (í ), CHENG Wei-Wen ( á ), LIU Tang-Kun ( ²), HUANG Yan-Xia (á ), and LI Hong (Ó ) College of Physics and Electronic Science, Hubei Normal University, Huangshi 35002, China (Received December 28, 2009) Abstract We investigate the sudden birth and sudden death of entanglement of two qubits interacting with uncorrelated structured reservoirs. The system is initially prepared in two-qubit extended Werner-like state. We work out the dependence of the entanglement dynamics on both non-markovian environments and the purity of initial state, and show that non-markovian environments and the purity can control the time of the two-qubit entanglement sudden death and the reservoirs entanglement sudden birth. Furthermore, under the conditions of different purity and initial entanglement, the revival of qubits entanglement can manifest before, simultaneously or even after the disentanglement of their corresponding reservoirs. PACS numbers: Ud, Mn Key words: entanglement dynamics, zero-temperature bosonic reservoir, non-markovian environments 1 Introduction Entanglement is a nonlocal correlation in quantum systems and plays a central role in the application of quantum information. [1 3] It has been extensively studied theoretically and creation of entanglement has now been reported in different systems, including cavity QED, [ 5] trapped ions, [6] spin systems, [7] atomic ensembles, [8] and photon pairs. [9] It has been shown that two entangled qubits can become completely disentangled in a finite time under the influence of pure vacuum noise. The phenomenon that is named as entanglement sudden death exhibits striking difference with the usual local decoherence, which has been theoretically predicted by Yu and Eberly, [10] and experimentally observed for entangled photon pairs [11] and atomic ensembles. [12] On the other hand, quantum systems suffer decoherence because of their interactions with the surrounding environment. The decoherence process is indeed the major obstacle for quantum information processing. Entanglement dynamics and decoherence have been studied in the frame of various models. [13 17] Typically, entanglement sudden death occurs when the two qubits interact with two independent environments. A completely different phenomenon appears when the qubits interact with their independent environments, respectively. Recently, a deeper understanding of the sudden death process has been gained by considering the quantum correlations shared by the environments, which show a sudden birth. In Ref. [18], the authors show that the ESD of a two-qubit system is intimately linked to entanglement sudden birth (ESB) of the independent reservoirs in the Markovian regime. From Ref. [19], the authors have taken Bell-like states as the initial state of atoms and assumed the initial cavity field to be in vacuum. As a consequence, ESD was found to be sensitive to the initial atomic state. That is to say, ESD may occur for a certain type of initial atomic state but does not appear for another type. In addition, due to the memory of the non-markovian environments, their fundamental importance in quantum information processing and quantum computation, non-markovian quantum dissipative systems have attracted much attention in recent years. Some studies have also showed that entanglement of qubits will revive in the case of a commonly shared reservoir [20 21] or of independent reservoirs. [22 2] Due to its great importance, here we focus on in detail the study of entanglement dynamics in non-markovian approximation for two qubits interacting, respectively, with two independent reservoirs. Different from the previous work, we prepare the initial state of the two qubits in the mixed extended Werner-like state, including pure and mixed states. Our results show that a full transfer of atomic entanglement into two reservoirs is possible in the non-markovian environments. The speed of the entanglement transfer and two-qubit entanglement revivals are related to the degree of entanglement of the initial states and the purity of initial state of two qubits. We also find the reservoirs entanglement can not only exhibit ESB but also ESD and entanglement revivals in the evolution. Furthermore, under the conditions of different purity and initial entanglement, the revival of atom entanglement is Supported by the National Natural Science Foundation of China under Grant No , Natural Science Foundation of Hubei Province under Grant No. 2009CDA15, Educational Commission of Hubei Province under Grant No. D and the Postgraduate Programme of Hubei Normal University under Grant No. 2007D20 scj1122@163.com tkliuhs@163.com

2 28 SHAN Chuan-Jia, LIU Ji-Bing, CHEN Tao, CHENG Wei-Wen, LIU Tang-Kun, HUANG Yan-Xia, and LI Hong Vol. 5 not always accompanied by the disentanglement of reservoirs, and vice versa, the revival of qubits entanglement can manifest before, simultaneously or even after the disentanglement of their corresponding reservoirs. Our results will be helpful in understanding the origin of the ESD/ESB and the relation between the ESD/ESB and different mixed portions in the initial states. 2 Theoretical Hamiltonian Model We consider an open quantum system consisting of two qubits A and B each coupled to a zero-temperature bosonic reservoir in the vacuum denoted a and b, respectively. In this paper, we will assume that each atomreservoir system is isolated and the reservoirs are initially in the vacuum state while the atoms are initially in extended Werner-like state. The interaction between an atom and an N-mode reservoir is described through the Hamiltonian under the dipole and the rotating-wave approximation and setting = 1. Ĥ = ωˆσ +ˆσ + ω jˆb jˆb j + g j (ˆσ ˆb j + ˆσ +ˆb j ), (1) where ˆb j, ˆb j are the creation and annihilation operators of the mode j of the reservoir, ˆσ + = 1 0, ˆσ = 0 1 and ω are the inversion operators and transition frequency of the qubit; ω j and g j are the frequency of the mode j of the reservoir and its coupling strength with the qubit. For an initial state of the form 1 0 r with 0 r = N 0 j r, then the evolution of the single qubit-reservoir system given by Eq. (1) leads to Φ(t) = C 0 (t) 1 0 r + C j (t) 0 1 j r, (2) where 1 j r is the state of the reservoir with only one exciton in the j-th mode. Setting δ j = ω ω j, the equations for the probability amplitudes take the form Ċ 0 (t) = i g j C j (t), (3) Ċ j (t) = igjc 0 (t). () Formally integrating Eq. () and inserting its solution into Eq. (3), we obtain an integro-differential equation for C 0 (t), M t Ċ 0 (t) = g j 2 dt 1 C 0 (t 1 ). (5) In the continuum limit for the reservoir spectrum the sum over the modes is replaced by the integral M g j 2 dωj(ω), where J(ω) is the reservoir spectral density. In the following we focus on the case, in which the structured reservoir is the electromagnetic field inside a lossy cavity. In 0 this case, the fundamental mode supported by the cavity displays a Lorentzian broadening due to the non-perfect reflectivity of the cavity mirrors. Then the spectrum of the field inside the cavity can be modeled as J(ω) = R2 λ π (ω ω c ) 2 + λ 2, (6) where the weight R is proportional to the vacuum Rabi frequency and λ is the width of the distribution and therefore describes the pseudomode decay rate into the reservoir, and ω c is the fundamental frequency of the cavity. Typically, according to Ref. [25], weak-coupling (λ > 2R), where the behavior of the qubit-reservoire system is Markovian and irreversible decay occurs, and strong-coupling regime (λ < 2R), where non-markovian dynamics occurs accompanied by an oscillatory reversible decay and a structured rather than a flat reservoir situation applies. In our paper, we will mainly make considerations to the non-markovian approximation, and show the different results from the Markovian treatment. Through introducing the correlation function f(t t 1 ) = dωj(ω)e i(ωc ω)(t t1) and performing the Laplace transform of Eq. (5), we acquire s C 0 (s) C 0 (0) = C 0 (s) f(s). (7) From the above equation we can derive the quantity C 0 (s). Finally, inverting the Laplace transform, we obtain a formal solution for the amplitude C 0 (t) = e λt/2[ ( Ωt ) cosh + λ ( Ωt )] 2 Ω sinh, λ > 2R, 2 = e λt/2[ ( Ωt ) cos + λ ( Ωt )] 2 Ω sin, λ < 2R, (8) 2 where Ω = λ 2 R 2. If we set C(t) = 1 C 0 (t) 2, Eq. (2) can be rewritten as with Φ(t) = C 0 (t) 1 0 r + C(t) 0 1 r, (9) 1 r = 1 C(t) M C j (t) 1 j r. For the initial state of qubit-pair AB, instead of Belllike and Werner states, [26] we shall consider the following extended Werner-like state ρ I AB(0) = r φ AB AB φ + 1 r I AB, (10) with r the purity of the initial state of qubits AB, I AB the identity matrix and φ AB = (cos θ 00 + sin θ 11 ) AB, (11) the Bell-like state. This class of mixed state arises naturally in a wide variety of physical situations. Obviously, the state in Eq. (11) reduces to the standard Werner state when θ = π/ and to Bell-like pure state when r = 1. By dealing with the above extended Werner-like state, we are able to study the effect of mixedness of the initial entangled state on the evolution of atoms and their corresponding reservoirs. The atom and its reservoir now evolve as an effective two-qubit system. We will study the

3 No. 3 Entanglement for Two-Qubit Extended Werner-Like States: Effect of Non-Markovian Environments 29 joint evolution of atoms A and B with their corresponding reservoirs a and b initially in the global state ρ I (0) ABab = ρ I AB 0 0 ab, (12) respectively. The evolution of the composite system reads ρ I (t) ABab = U(t) ρ I AB 0 0 ab U (t), (13) where the coefficients for the states ρ I (t) ABab are given by Eq. (8). 3 Entanglement Dynamics In order to describe the entanglement dynamics of the bipartite system, we use the concurrence, which is proposed by Wootters [26] as a measure of entanglement. The concurrence C = 0 corresponds to a separable state and C = 1 to a maximally entangled state. Nonzero concurrence means that the two qubits are entangled. For a system described by the above density matrix, which can denote either a pure or a mixed state, the concurrence is defined as C(ρ) = max(0, λ 1 λ 2 λ 3 λ ), (1) where λ 1, λ 2, λ 3, λ are the eigenvalues in a decreasing order of the spin-flipped density operator R defined by R = ρ ρ ρ with ρ = (σy σ y )ρ (σ y σ y ), ρ denotes the complex conjugate of ρ, σ y is the usual Pauli matrix. For Both the Bell-like state and Werner state, and so the extended Werner-like state, belong to the so-called X- class state whose density matrix is of the form x 0 0 v 0 y u 0 ρ = 0 u z 0, (15) v 0 0 w with x, y, z, w real positive and u, v complex quantities. For the X-state equation (15), the concurrence can be derived as C(ρ) = 2 max{0, u xw, v yz}. (16) Firstly, the total system state at t = 0 is ρ I (0) ABab. After time t, the reduced density matrix ρ I (t) ABab can be obtained by tracing over the degrees of freedom of qubits a and b, which remains the X-form with x = 1 r + r(cos 2 θ + sin 2 θ χ A 2 χ B 2 ), y = 1 r + r sin 2 θ χ A 2 ξ B 2, z = 1 r + r sin 2 θ ξ A 2 χ B 2, w = 1 r + r sin 2 θ ξ A 2 ξ B 2 ), u = 0, v = r sin θ cosθξa ξ B. (17) And the reduced density matrix form of ρ φ ab (t) is the same as ρ φ AB(t) but with the following matrix elements x = 1 r + r(cos 2 θ + sin 2 θ ξ A 2 ξ B 2 ), y = 1 r + r sin 2 θ ξ A 2 χ B 2, z = 1 r + r sin 2 θ χ A 2 ξ B 2, w = 1 r + r sin 2 θ χ A 2 χ B 2, u = 0, v = r sinθ cosθχ A χ B. (18) The functions ξ j (t), χ j (t) are given by ξ j (t) = e λjt/2[ cos(ω j t/2) + λ ] j sin(ω j t/2), Ω j (19) χ j (t) = 1 ξ j (t) 2, (20) where j = A, B. For simplicity, λ A = λ B = λ, Ω A = Ω B = Ω. By virtue of Eq. (16), we can get the concurrences C(ρ φ AB(t)) and C(ρ φ ab (t)) as C φ AB(ab) = 2 max{0, v yz}. (21) In this section we will analyze the entanglement evolution in the double atom-reservoir model. Our aim is to investigate the effects of the the mixedness of the initial states and the pseudomode decay rate on the entanglement transfer between atomic subsystem and reservoir subsystem. The two-qubit entanglement dynamics has been previously analyzed taking pure Bell-like states as initial states. However, as said before, it appears of interest to study the entanglement dynamics of two independent qubits, each locally interacting with a reservoir, in the case of more general initial conditions and in particular for Werner states. The procedure we have developed is applicable to any two-qubit initial state and therefore also to extended Werner-like states. The dynamics of bipartite entanglement between the qubits of our model show interesting features. Figures 1 and 2 show the time evolution of concurrences between the two atoms (C(ρ φ AB(t))) and the two reservoirs (C(ρ φ ab (t))) for various values of the parameter θ with pure Bell-like states as initial states (r = 1) in the non-markovian regime (λ = 0.1R). This choice of λ = 0.1R corresponds to an experimentally feasible value in the cavity QED context. For various values of the parameter θ, corresponding to the two qubits are initially in different entangled states, we find that under some initial conditions, the entanglement between the two atoms occurs sudden death, sudden birth of entanglement arises between the two reservoirs. According to the memory effect of the non-markovian reservoirs, two atoms entanglement can revive after their entanglement sudden death, and reservoirs entanglement can present sudden birth and death phenomenon; while under some initial conditions, the atoms entanglement can not disappear after a finite time without entanglement sudden death of two atoms, and the reservoirs entanglement sudden birth does not

4 30 SHAN Chuan-Jia, LIU Ji-Bing, CHEN Tao, CHENG Wei-Wen, LIU Tang-Kun, HUANG Yan-Xia, and LI Hong Vol. 5 occur. That is to say, we can prepare certain initial entanglement states to prolong entanglement time. when the impurity is very low, the effects of impurity are not strong enough to weaken the entanglement, so continuous entanglement evolution appears. But the higher impurity the stronger the effects of impurity on the entanglement evolution, as a result, entanglement sudden death and birth become more obvious with increasing impurity. The other is that for intermediate values of purity r, entanglement revivals after finite dark periods occur for the initial Werner-like states. When r < r c, the entanglement of two atoms and two reservoirs will remain zero, which can be seen in Figs. 3 and. That is to say, entanglement and entanglement transfer exist for these states when r > r c. Fig. 1 Time evolution of concurrences between the two atoms (C(ρ φ AB(t))) for various values of the parameter θ with pure Bell-like states as initial states (r = 1) in the non-markovian regime (λ = 0.1R). Fig. 3 Evolution of two-qubit concurrences C AB as a function the dimensionless parameter r with θ = π/ and λ = 0.1R. Fig. 2 Time evolution of concurrences between the two reservoirs (C(ρ φ ab (t))) for various values of the parameter θ with pure Bell-like states as initial states (r = 1) in the non-markovian regime (λ = 0.1R). In this part, we will focus on the entanglement dynamical evolution of the composite system in extended Werner-like states, i.e., for r 1. The choice of θ corresponds to the Bell-like initial states, while the choice of the purity rests on the fact that in this condition the initial state for the atoms is more realistic. In Figs. 3 and, we show the concurrence as a function of both t and the dimensionless parameter r. The results show that the ESD of two atoms and ESB of two cavities occur in some situations but at different times, the differences depend on the values of purity r. In addition, one can also observe that there are two interesting features in the entanglement evolution. One is that the pairwise entanglement oscillates with time continuously at the high purity, but as the impurity increases the entanglement evolves to zero and will remain zero for a period of time before the entanglement recovers. In particular, increasing the purity of the initial states, the ESD time is retarded for non-markovian environments. A physical interpretation of the result is that Fig. Evolution of two-reservoir concurrences C ab as a function the dimensionless parameter r with θ = π/ and λ = 0.1R. Another aspect of interest is how the entanglement dynamics and entanglement transfer are influenced by the values of λ and r. Due to the interactions in lossy cavities, the atomic entanglement can disappear eventually, and the entanglement between the two reservoirs can reach a stationary value after a certain time, which is equal to the initial entanglement of the two atoms. That is to say, the AB subsystem s entanglement is transferred to the ab system s entanglement thoroughly.

5 No. 3 Entanglement for Two-Qubit Extended Werner-Like States: Effect of Non-Markovian Environments 31 Fig. 5 Evolution of two-qubit concurrences C AB (solid line) and C ab (dotted line), for different parameter r with θ = π/, λ = 0.1R. Fig. 6 Evolution of two-qubit concurrences C AB (solid line) and C ab (dotted line), for different parameter λ with θ = π/, r = 0.8. From Fig. 5, we can see that entanglement sudden death and sudden birth can appear much more easily in initial Werner-like states than initial Bell-like states with the increase of the initial states mixedness. As we show, concurrence actually goes abruptly to zero in a finite time and remains zero thereafter. That is to say, the entanglement sudden death always happens in the strong initial states mixedness. Meanwhile, the revival of atom entanglement is not always accompanied by the disentanglement of reservoirs, and vice versa. Entanglement of two atoms can revive before, simultaneously or even after the ESD of two reservoirs and these phenomena are dependent of the relative strength of initial states mixedness. When we consider the case that the different pseudomode decay λ is in each lossy cavity, such as λ = 0, 0.2, 0.5R (also satisfy the condition of non-markovian effects). From Fig. 6, we find that in the case of λ = 0.1R, one can achieve complete transfer of an initial entanglement to the reservoirs at some special time instant and entanglement sudden death and sudden birth can occur quickly in the larger pseudomode decay cavities, and the AB subsystem s entanglement is transferred to the ab system s entanglement thoroughly after the time, which is much smaller than that in the case λ = 0.1R. When the atomic entanglement disappears thoroughly, it is clearly shown that the reservoirs entanglement arrives to the value of the two atoms initial entanglement in the long time limit. That is because the atomic entanglement is also transferred to the Ab, Ba, Aa, and Bb subsystems during the time evolution of the two-qubit entanglement, and when these entanglements disappear completely, the reservoirs entanglement would reach the value of the two atoms initial entanglement. The initial entanglement of atoms shall be transferred into the reservoirs in the strong pseudomode decay for non-markovian environments. That is to say, in the

6 32 SHAN Chuan-Jia, LIU Ji-Bing, CHEN Tao, CHENG Wei-Wen, LIU Tang-Kun, HUANG Yan-Xia, and LI Hong Vol. 5 lossy cavities system, it is interesting to find that the initial atomic entanglement with Werner-like states can be totally transferred to the two reservoirs system. Conclusions To summarize, we investigate the sudden birth and sudden death of entanglement of two qubits, which are prepared in two-qubit extended Werner-like states and interact with uncorrelated structured reservoirs. Under the conditions of non-markovian environments lossy cavities and the purity, the atomic entanglement is transferred to the reservoirs entanglement thoroughly after one time instant, the larger pseudomode decay can make this entanglement transfer more easily. We also find that non- Markovian environments and the purity can control the time of the two-qubit entanglement sudden death and the reservoirs entanglement sudden birth. Furthermore, under the conditions of different purity and initial entanglement, the revival of qubits entanglement can manifest before, simultaneously or even after the disentanglement of their corresponding reservoirs. We believe that our results contribute in shedding light on the behavior of quantum entanglement in realistic conditions, that is when the effects of the non-markovian environment and the purity on the quantum system are taken into account. The experimental implementation seems to be feasible due to the recent advances in deterministic trapping of atoms in the optical cavities. [27 28] In physical contexts, the observation of the effects we have discussed should be achievable with the current experimental technologies. References [1] M.A. Nielsen and I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge (2000). [2] Z.Y. Xue, Y.M. Yi, and Z.L. Cao, Physica A 37 (2007) 119; Z.Y. Xue, Y.M. Yi, and Z.L. Cao, J. Mod. Opt. 53 (2006) [3] L. Grover, Phys. Rev. Lett. 80 (1998) 329 [] S.B. Zheng and G.C. Guo, Phys. Rev. Lett. 85 (2000) 2392; G. Zhang, M. Yang, and Z.L. Cao, Commun. Theor. Phys. 9 (2008) 117. [5] C.J. Shan, W.W. Cheng, T.K. Liu, D.J. Guo, and Y.J. Xia, Commun. Theor. Phys. 9 (2008) 1505; C.J. Shan and Y.J. Xia, Acta Phys. Sin. 55 (2006) [6] Q.A. Turchette, C.S. Wood, B.E. King, C.J. Myatt, D. Leibfried, W.M. Itano, C. Monroe, and D.J. Wineland, Phys. Rev. Lett. 81 (1998) [7] X.G. Wang, Phys. Rev. A 6 (2001) ; ibid. 66 (2001) 0305; ibid. 66 (2001) [8] B. Julsgaard, A. Kozhekin, and E.S. Polzik, Nature (London) 13 (2001) 00. [9] A. Aspect, P. Grangier, and G. Roger, Phys. Rev. Lett. 7 (1981) 60. [10] T. Yu and J.H. Eberly, Phys. Rev. Lett. 93 (200) 100. [11] L. Aolita, R. Chares, D. Cavalcanti, A. Acín, and L. Davidovich, Phys. Rev. Lett. 100 (2008) [12] M.P. Almeida, F.de Melo, M. Hor-Meyll, A. Salles, S.P. Walborn, P.H. Souto Ribeiro, and L. Davidovich, Science 316 (2007) 579. [13] Z. Ficek and R. Tanaś, Phys. Rev. A 7 (2006) 0230; Z.X. Man, Y.J. Xia, and B.A. Nguyen, J. Phys. B 1 (2008) ; Z.X. Man, Y.J. Xia, and B.A. Nguyen, J. Phys. B 1 (2008) ; X.X. Yi and W. Wang, Phys. Rev. A 76 (2007) [1] T. Yu and J.H. Eberly, Phys. Rev. Lett. 97 (2006) 1003; J.H. Eberly and T. Yu, Science 316 (2007) 555; T. Yu and J.H. Eberly, Quantum Inf. Comput. 7 (2007) 59. [15] M.P. Almeida, F.de Melo, M. Hor-Meyll, A. Salles, S.P. Walborn, P.H. Souto Ribeiro, and L. Davidovich, Science 316 (2007) 579. [16] Y.J. Zhang, Z.X. Man, and Y.J. Xia, J. Phys. B 2 (2009) [17] J. Liu, Z.Y. Chen, S.P. Bu, and G.F. Zhang, Commun. Theor. Phys. 52 (2009) 133; Z.Y. Chen and G.F. Zhang, Opt. Commun. 275 (2007) 27 [18] C.E. López, G. Romero, F. Lastra, E. Solano, and J.C. Retamal, Phys. Rev. Lett. 101 (2008) [19] M. Yönaç, T. Yu, and J.H. Eberly, J. Phys. B 0 (2007) S5. [20] Z. Ficek and R. Tanaś, Phys. Rev. A 7 (2006) 0230; Z. Ficek and R. Tanaś, Phys. Rev. A 77 (2008) [21] S. Maniscalco, F. Francia, R.L. Zaffino, N.L. Gullo, and F. Plastina, Phys. Rev. Lett. 100 (2008) ; Y. Li, J. Zhou, and H. Guo, Phys. Rev. A 79 (2009) [22] B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. Lett. 99 (2007) ; B. Bellomo, R. Lo Franco, and G. Compagno, Phys. Rev. A 77 (2008) [23] J. Dajka, M. Mierzejewski, and J. Luczka, Phys. Rev. A 77 (2008) [2] Xiu-Feng Cao and Hang Zheng, Phys. Rev. A 77 (2008) [25] B.J. Dalton, S.M. Barnett, and B.M. Garraway, Phys. Rev. A 6 (2001) [26] W.K. Wootters, Phys. Rev. Lett. 80 (1998) 225. [27] A.D. Boozer, A. Boca, R. Miller, T.E. Northup, and H.J. Kimble, Phys. Rev. Lett. 97 (2006) [28] K.M. Fortier, S.Y. Kim, M.J. Gibbons, P. Ahmadi, and M.S. Chapman, Phys. Rev. Lett. 98 (2007)