Stationary quantum correlations in Tavis Cumming model induced by continuous dephasing process

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1 Quantum Inf Process (03) : DOI 0.007/s Stationary quantum correlations in Tavis Cumming model induced by continuous dephasing process Wei Wu Hang-Shi Xu Zheng-Da Hu Jing-Bo Xu Received: 5 April 03 / Accepted: May 03 / Published online: 3 May 03 Springer Science+Business Media New York 03 Abstract We investigate the dynamics of quantum correlations for the Tavis Comming model in continuous dephasing process. It is shown that quantum discord and entanglement can reach stationary value, and quantum Zeno effect occurs in strong-coupling region. Furthermore, we explore the influence of continuous dephasing process on the trace distance between two marginal states of the two atoms and find that the trace distance also achieve a constant value during time evolution. Keywords Continuous dephasing process Quantum correlations Trace distance Introduction The problem of quantum decoherence which is associated with many interesting phenomena has attracted considerable attention for long time [,]. The decoherence process help us to predict the results of experiments as well as to point out working conditions where genuine quantum phenomena are expected [3 5]. A environmentinduced decoherence system, which is open and interacts with the environment schematizing the meter, can be described by master equation in the Lindblad form which is the evolution equation of reduced density operator influenced by environment [6,7]. Different kinds of decoherence environment, which are represented by different operators and depend upon the specific experiments, have been extensively investigated [8,9]. Quantum entanglement which was first introduced by Einstein, Podolsky and Rosen in their famous paper in 935 [0] has become a hot topic in quantum information W. Wu H.-S. Xu Z.-D. Hu J.-B. Xu (B) Zhejiang Institute of Modern Physics and Physics Department, Zhejiang University, Hangzhou 3007, China xujb@zju.edu.cn

2 39 W. Wu et al. and quantum computation. It has been recognized that entanglement which is a special quantum correlation can be used as an important resource for quantum information processing []. The experimental demonstrations of two-particle entanglement and multi-particle entanglement in the cavity quantum electrodynamics have been reported [,3] and some applications about the entanglement or the nonclassical correlations have also been realized in recent experiments [4 6]. However, some recent studies show that there are other kinds of quantum correlations that can offer support for the quantum tasks. It has been proved both theoretically [7,8] and experimentally [9] that other nonclassical correlation, namely, quantum discord [0] can be responsible for quantum computation. Quantum discord, which is defined as the difference between the quantum mutual information and the classical correlation, is nonzero even for separate mixed states and may be regarded as a more general and fundamental resource in the quantum information processing. It is well known that quantifying the distinguishability of quantum states precisely is of importance in quantum information processing []. One of measures for the distinguishability of two quantum states is trace distance. Trace distance is a natural metric on the space of physical states, satisfying several attractive properties which are desirable for the measure of distance between two states []. Recently, it has been theocratically reported [3,4] and experimentally identified [5] that the trace distance has a close relation with the non-markovian behavior of open quantum systems. In this paper, we study the dynamical evolution of Tavis Comming model [6, 7] in continuous dephasing process. We solve the master equation and obtain an explicit expression for the initial state which two atoms are coupled to a common vacuum field. By making use of the explicit expression we explore the influence of continuous dephasing process on the dynamics of quantum correlations between two atoms. It is found that the quantum discord and entanglement can reach stationary value, and quantum Zeno effect [8 30] occurs in strong-coupling region. The values of stationary quantum discord and entanglement can be controlled by adjusting the parameters which are related to the Hamiltonian and initial state. Moreover, we explore the influence of continuous dephasing processs on the trace distance between two marginal states of the two atoms and find that the trace distance also achieve a constance value during time evolution. Finally, we adopt Monte Carlo wave function method [3] to numerically investigate dynamics of the entanglement and discord between two atoms which are coupled to excited field in continuous dephasing process. Tavis Cumming model in continuous dephasing process We consider the Tavis Comming model consisting of two identical two-level atoms A and B which do not interact directly and are coupled to a single-model cavity field. The Hamiltonian is given by ( h = ) H = ωa a + ω 0 (σ z A + σ z B ) + j=a,b λ j (aσ + j + a σ j ), ()

3 Stationary quantum correlations 393 where a and a denote the annihilation and creation operators of the single-model cavity field with frequency ω, σj z and σ j ± are the atomic operators, and ω 0 is the atomic transition frequency. λ j is the coupling constants between the cavity field and atom j. By tracing over the degree of freedom of the external environment, the master equation for the reduced density operator ρ of the system can be written in the Lindblad form ρ = i[h,ρ] κ [A, [A,ρ]], () where the parameter κ denotes the coupling strength of the system to the meter, H is the Hamiltonian of closed system and the Lindblad operator A = a a represents the influence of the environment which is related to the measured observable. To solve the master equation, we first construct a conserved quantity, K = a a + σ z A + σ B z +. (3) It is easy to verify that K commutes with the Hamiltonian H. Then, four eigenvectors ψ n (i) which satisfies K ψ n (i) =n ψ n (i) and H ψ n (i) =E n (i) ψ n (i) can be expressed in the subspace K = n as ψ n (i) = 4 i= α i i n, where α i is a complex number, and n = ggn, n = gen, 3 n = egn, 4 n = een. For simplicity, we consider the resonant case ω = ω 0 and n =. Then, the eigenvectors and eigenvalues of Humilation can be calculated as ψ () = gg0, (4a) ψ () = λ A λ ge0 +λ B λ eg0, (4b) ψ (3) = λ B ge0 + λ A eg0 + gg, λ λ (4c) ψ (4) = λ B ge0 + λ A eg0 gg. λ λ (4d) E () = ω; E () = 0, (5a) E (3) = E (4) = λ. (5b) where λ = λ A + λ B. Next, we solve the master equation exactly in the basis { ψ (i) }, and obtain differential equations as follows

4 394 W. Wu et al. ρ 33 = κ 4 (ρ 44 ρ 33 ), ρ 44 = κ 4 (ρ 33 ρ 44 ), ( ρ 34 = iλ κ ) ρ 34 + κ 4 4 ρ 43, ρ 43 = ( iλ κ 4 ) ρ 43 + κ 4 ρ 34. (6a) (6b) (6c) (6d) with ρ ij = ψ (i) j) ρ ψ( and the rest ρ ij = 0. We consider that the initial state of the two atoms is prepared in ρ S (0) = ϕ AB ϕ AB, (7) where ϕ AB =(cos θ ge +sin θ eg ) AB, and the cavity is initially in the Fock-state m. The initial state for the total system is then given by ρ(0) = ϕ AB ϕ AB m m. (8) For the initial state with m = 0, we can obtain analytic solutions for the differential equations as follows ρ 33 (t) = ρ 44 (t) = (λ A sin θ + λ B cos θ) λ, (9) ρ 34 (t) = ρ43 (t) = (λ A sin θ + λ B cos θ) λ e κ 4 t ξ [ ) ξ cosh + (κ + 8iλ) sinh ( 4 ξt ( )] 4 ξt. (0) where ξ = κ 64λ. () For the general initial state case with m, we use the Monte Carlo wave function method to solve the master equation and obtain numerical simulations shown in Sect. 5. It is easy to see that ρ 33 and ρ 44 depend only on λ i and initial state parameter θ. In the limit t + (stationary state situation), we obtain The density matrix ρ(t) is given by ρ 43 (+ ) = ρ 34 (+ ) = 0. () ρ(t) = ij ρ ij (t) ψ (i) j) ψ(, (3)

5 Stationary quantum correlations 395 and the reduced density matrix ρ S = Tr E (ρ) is expressed in the basis { gg, ge, eg, gg }as where τ 000 ρ S = 0 δη0 0 ηɛ0, (4) 0000 τ = (ρ 33 + ρ 44 ρ 34 ρ 43 ), (5) δ = λ B λ (ρ 33 + ρ 44 + ρ 34 + ρ 43 ), (6) η = λ Aλ B λ (ρ 33 + ρ 44 + ρ 34 + ρ 43 ), (7) ɛ = λ A λ (ρ 33 + ρ 44 + ρ 34 + ρ 43 ). (8) 3 The influence of continuous dephasing process on the quantum discord In this section, we explore the influence of both weak-coupling and strong-coupling cases on dynamics of quantum discord between two atoms. First, we briefly outline the concepts of quantum discord between a bipartite system (say qubits A and B). The quantum discord which is defined as the difference between the quantum mutual information I(ϱ AB ) and the classical correlation C(ϱ AB ) can be expressed as Q(ϱ AB ) = I(ϱ AB ) C(ϱ AB ), (9) I(ϱ AB ) = S(ϱ A ) + S(ϱ B ) S(ϱ AB ), (0) where ϱ A(B) and ϱ AB are the reduced density matrix of subsystem A(B) and the density matrix of total system, respectively, and S(ϱ) = Trϱ log ϱ is the von Neumann entropy. The classical correlations of the state C(ϱ AB ) are given by C(ϱ AB ) = max {B k } {S(ϱ A) S(ϱ AB {B k })}, () where {B k } is a complete set of projectors preformed on subsystem B locally. S(ϱ AB {B k }) = k p ks(ϱ k ) is the based-on-measurement quantum conditional entropy with ϱ k = Tr B [(I B k )ϱ AB (I B k )]/p k and the probability p k = Tr AB [(I B k )ϱ AB (I B k )]. By substituting Eqs. () and (0) into Eq. (9), we obtain the quantum discord as Q(ϱ AB ) = S(ϱ B ) S(ϱ AB ) + min {B k } [ k p k S(ϱ k )]. ()

6 396 W. Wu et al. For an X-state in Eq. (39), S(ϱ AB ) is given by [3], 3 S(ϱ AB ) = χ l log χ l, (3) l=0 where χ 0, = [(a + d) ± ] (a d) + 4 ω, (4) χ,3 = [(b + c) ± ] (b c) + 4 z, (5) are the four eigenvalues of ϱ AB, and where S(ϱ B ) = H(a + c), (6) H(x) = [ x log x + ( x) log ( x) ], (7) is the Shannon entropy. In order to derive the classical correlation can be obtained by using the similar procedure, a complete set of projective measurements performed locally on subsystem B is introduced, i.e., {B k = W k W } where { k = k k, k = g, e} is the set of projectors on the computational basis ( g = and e = ) and W is a unitary operator. Since B k represents any von Neumann projective measurement, the unitary operator W can have the matrix form as W = ( cos β e iφ sin β ) e iφ sin β cos β. (8) where 0 β π, 0 φ<π. After the measurement we obtain where p k = [ + ( ) k (a b + c d) cos β ( + γ k S(ϱ k ) = H ], (9) ), (30) u γ k = + v + ( ) k [(a + c) ] cos β, (3) u = ( ) k [(a + b) ]+[(a + d) ] cos θ, (3) v = (ω + z) sin β. (33)

7 Stationary quantum correlations 397 Fig. a The dynamics of quantum discord without continuous dephasing process with θ = π 3 for different λ i : λ A = λ B = (blue dashed line), λ A = λ B = (red dot dashed line), λ A = λ B = 3(black solid line). b The dynamics of discord in continuous dephasing process with parameter κ = 5, other parameters are the same with (Color figure online) Then min {Bk }[ k p k S(ϱ k )] can be obtained at β = 0orβ = π min [ k p k S(ϱ k )]=min[c, C ], (34) {B k } where ( ) ( ) c b C = (a + c)h + (b + d)h, (35) a + c b + d ( + ) (a + b c d) C = H + 4(ω + z). (36) In Fig. a, we plot the dynamical evolution of quantum discord without continuous dephasing process, i.e., κ = 0. It can be seen clearly from the figure that the quantum discord of system periodically oscillates without continuous dephasing process. In continuous dephasing process, i.e., κ = 0, we first consider the weak-coupling case. We displayed the quantum discord as a function of time t for different λ A and λ B with fixed κ = 5 and θ = π 3 in Fig. b. We observe that the quantum discord of two atoms still oscillates at the beginning and then reaches a stationary state quantum discord finally. It can be observed from Fig. a that κ affects the time when system reaches the stationary state entanglement, with the increase of κ, the time to reach stationary state quantum discord become shorter. The value of stationary quantum discord can be controlled by adjusting the parameters λ i as displayed in Fig. b. In the strong-coupling case, we plot the quantum discord as a function of time t for different values of κ with fixed λ A = λ B = and θ = π 3 in Fig. 3. It is quite clear to see that the quantum discord directly reaches the stationary value without oscillates. In contrast with Fig. b, in the limit κ +, the quantum discord remain unchanged at its initial value, which is the quantum Zeno effect [8 30].

8 398 W. Wu et al. Fig. a The dynamics of quantum discord with different coupling parameter κ = (blue dashed line), κ = (red dot dashed line), κ = 5(black solid line), other parameters are the same: λ A =,λ B = and θ = π 3. b The dynamics of quantum discord with different parameter λ B : λ B = (blue dashed line), λ B = (red dot dashed line), λ B = 3(black solid line), other parameters are the same: λ A =,κ = 5 and θ = π 3 (Color figure online) Fig. 3 The time evolution of discord in strong-coupling case with fixed λ A = λ B = and θ = π 3 for different parameter κ: κ = 0 (orange solid line), κ = 30 (blue dot dashed line), κ = 50 (red dashed line)and κ = 0000 (black dotted line) (Color figure online) 4 The influence of continuous dephasing process on entanglement In the section, we explore the dynamics of quantum entanglement between two atoms in continuous dephasing process. In order to quantify the degree of entanglement and make a comparison with quantum discord, we adopt the measure of concurrence [33] to calculate the entanglement between two atoms. The concurrence related to the density operator ϱ of a mixed state is defined by C = max {0, λ λ λ 3 } λ 4, (37) where λ i are the eigenvalues in decreasing order of the matrix ϱ(σ y σ y )ϱ (σ y σ y ). (38) ϱ denotes the complex conjugation of ϱ and σ y is the standard Pauli matrix. The concurrence varies from C = 0 for an unentangled state to C = for a maximally entangled state.

9 Stationary quantum correlations 399 Fig. 4 a The dynamics of entanglement without continuous dephasing process with fixed θ = π 3 for different λ i : λ A = λ B = (blue dashed line), λ A = λ B = (red dot dashed line), λ A = λ B = 3(black solid line) b The dynamics of entanglement in continuous dephasing process with parameter κ = 5, other parameters are the same with (Color figure online) If ϱ is an X state a 0 0 ω ϱ X = 0 b z 0 0 z c 0, (39) ω 0 0 d which arises in a wide variety of physical situations, the concurrence can be easily derived as [34], { C(ϱ X ) = max 0, z ad, ω } bc. (40) By making use of Eqs. (4 8), we derive the concurrence as λ A λ B (ρ 33 + ρ 44 + ρ 34 + ρ 43 ) C = λ A + λ. (4) B In Figs. 4 and 6, we plot the dynamical evolution of concurrence in both weakcoupling and strong-coupling case respectively, the results are similar to the situation of quantum discord. The influence of parameter κ and λ i on the dynamics of concurrence is also explored (see Fig. 5): the value of κ affects the time when system reach stationary state situation and the values of the stationary state concurrence are sensitive to different combinations of λ A and λ B s values, moreover, the stationary state concurrence reaches maximum value when λ A = λ B with fixed θ. In particular, as shown in Fig. 6, the quantum Zeno effect occur under the limit κ, i.e., the value of entanglement can be remain unchanged in initial value. When t +, we obtain the stationary value of concurrence from Eqs. (7) and (4), C = λ A λ B (λ A sin θ + λ B cos θ), (4) λ 4

10 300 W. Wu et al. Fig. 5 a The dynamics of entanglement with different coupling parameter κ = (blue dashed line), κ = (red dot dashed line), κ = 5(black solid line), other parameters are the same: λ A =,λ B = andθ = π 3. b The dynamics of entanglement with different parameter λ B : λ B = (blue dashed line), λ B = (red dot dashed line), λ B = 3(black solid line), other parameters are the same: λ A =,κ = 5andθ = π 3 (Color figure online) Fig. 6 The time evolution of entanglement in strong-coupling case with fixed λ A = λ B = and θ = π 3 for different parameter κ : κ = 0 (orange solid line), κ = 30 (blue dot dashed line), κ = 50 (red dashed line)andκ = 0000 (black dotted line) (Color figure online) in terms of which the maximum of stationary state concurrence is C max = 0.5 if λ A = λ B and θ = π 4, while θ = 0 and λ i = 0, we can see C = 0, which means there still exist stationary state quantum entanglement even the two atoms initially in a separable state. 5 The influence of continuous dephasing process on trace distance Next we discuss the influence of continuous dephasing process on the time evolution of trace distance. The trace distance which can be used as a metric on the space of physical states is defined as ( D ρ,ρ ) = ρ Tr ρ. (43) where A = A A is the positive square root eigenvalues of A A. The trace distance D yields a natural metric on the state space and satisfies 0 D with D(ρ,ρ ) = 0 if and only if ρ = ρ and D(ρ,ρ ) = if and only if ρ is orthogonal to ρ.it has many properties that make it a useful measure for the distance between quantum states. First, the trace distance is preserved under unitary transformations U,

11 Stationary quantum correlations 30 Fig. 7 a The dynamics of trace distance between ρ S (0) and ρ S (t) without continuous dephasing process with θ = π 3 for different λ i : λ A = λ B = (blue dashed line), λ A = λ B = (red dot dashed line), λ A = λ B = 3(black solid line) b The dynamics of trace distance between ρ S (0) and ρ S (t) in continuous dephasing process with parameter κ = 5, other parameter are the same with (Color figure online) ( D Uρ U, Uρ U ) ( = D ρ,ρ ). (44) Second, the trace distance possesss the contractivity property ( D ρ, ρ ) ( D ρ,ρ ), (45) where is positive and trace preserving quantum operation. Third, the trace distance is subadditive ( D ρ σ,ρ σ ) ( D ρ,ρ ) ( + D σ,σ ). (46) and convex D j p j ρ j, j p j ρ j j ( p j D ρ,ρ ). (47) with p j satisfying j p j =. The trace distance between ρ S (0) and ρ S (t) as function of time t without continuous dephasing process is plotted in Fig. 7a. It is clear to see trace distance decays and revives periodically as time increases and the revival period is inversely proportional to the value of λ. We also plot the dynamics of trace distance in continuous dephasing process in Fig. 7b, a constant value of trace distance can be obtained as time increases. We study the influence of coupling strength κ on the dynamics of trace distance and find that with κ increases, the time at which trace distance reaches constant value become shorter as displayed in Fig. 8a. Trace distance as function of time t for different λ B (fixed λ A ) is shown in Fig. 8b, we observe the value of constant increases as parameter λ B increases which mean constant value of trace distance can be controlled by changing λ i. The case of strong-coupling case is also studied (see Fig. 9), we find the dynamics of trace distance reaches constant value without decays and revives when κ is large.

12 30 W. Wu et al. Fig. 8 a The dynamics of trace distance with different coupling parameter κ = (blue dashed line), κ = (red dot dashed line), κ = 5(black solid line), other parameters are the same: λ A =,λ B = andθ = π 3. b The dynamics of trace distance with different parameter λ B : λ B = (blue dashed line), λ B = (red dot dashed line), λ B = 3(black solid line), other parameters are the same: λ A =, κ= 5andθ = π 3 (Color figure online) Fig. 9 The time evolution of trace distance in strong-coupling case with fixed λ A = λ B = and θ = π 3 for different parameter κ : κ = 0 (orange solid line), κ = 00 (blue dot dashed line), κ = 500 (red dashed line) andκ = 0000 (black dotted line)(colorfigure online) In particular, the value of trace distance remain unchanged at its initial value under the limit κ, which is the quantum Zeno effect [8 30]. 6 Monte carlo wave function method for general initial states In the previous sections, we have investigated the influence of continuous dephasing process on quantum discord and entanglement for the initial state as ρ(0) = ϕ AB ϕ AB 0 0. For a more general initial state case m, there is no exact expressions,thus we adopt Monte Carlo wave function method [3] to numerically investigate dynamics of the quantum discord and entanglement for the general initial states with m. Without loss of generality, we choose the initial state as follows ρ(0) = ϕ AB ϕ AB m = m =. (48) The results of numerical simulations are showed in Figs. 0, and where we plot the concurrence and quantum discord as a function of time t in both weakcoupling and strong-coupling cases. It can be seen from the figures that the numerical simulations are similar to the analytic solution case. For the initial state in Eq. (48),

13 Stationary quantum correlations 303 Fig. 0 The entanglement as a function of time t in weak-coupling region. a The time evolution of entanglemment with κ = andθ = π 4 for λ A = λ B = 5; b The time evolution of entanglement with κ =.8andθ = π 4 for λ A = λ B = 5 Fig. The quantum discord as a function of time t in weak-coupling region. a The time evolution of quantum discord with κ = andθ = π 4 for λ A = λ B = 5; b The time evolution of quantum discord with κ =.8andθ = π 4 for λ A = λ B = 5 Fig. a The time evolution of entanglement in strong-coupling region with fixed λ A = λ B = 3and θ = π 4 for κ = 35 (blue solid line) andκ = 0000 (black dashed line); b The time evolution of quantum discord in strong-coupling region with fixed λ A = λ B = 3andθ = π 4 for κ = 35 (Solid line) and κ = 0000 (black dashed line) (Color figure online) continuous dephasing process can also lead stationary entanglement (discord) and with κ increases, the time at which system reach the stationary values of quantum correlations. From Fig., we can see that the quantum Zeno effect also occurs when

14 304 W. Wu et al. κ +, i.e., the value of concurrence and quantum discord remain unchanged and the stationary values of quantum correlations is different from the case m = 0. 7 Conclusions In conclusion, we adopt both analytic and numerical simulation methods to investigate the dynamical evolution of Tavis Comming model in continuous dephasing process. For the initial state ρ(0) = ϕ AB ϕ AB 0 0, we obtain exact expressions for the master equation of the system. And a more general initial state ρ(0) = ϕ AB ϕ AB m m where m, we adopt Monte carlo wave function method to obtain numerical simulations. Using these analytic results we have discussed the influence of continuous dephasing process on the quantum discord and entanglement between two atoms. It is found that quantum discord and entanglement can reach stationary value under the influence of continuous dephasing process. In the case of weak coupling region, the time evolutions of quantum discord and entanglement oscillate at the beginning and then reach a stationary value finally. By contrast, in the strong coupling case, quantum discord and entanglement reach stationary state situation without any oscillation. In the limit κ +, the quantum Zeno effect occurs and the stationary values of quantum discord and entanglement remain unchanged. Furthermore, the value of κ increases, the time at which the quantum discord and entanglement reach stationary values become shorter and the stationary values can be controlled by adjusting the parameters which are related to the Hamiltonian and initial state. There exits stationary quantum entanglement even for two atoms initially in a separable state. Moreover, The influence of continuous dephasing process on the trace distance between ρ S (0) and ρ S (t) is also explored. We find that the trace distance can achieve a constant value during time evolution in weak coupling region and exhibit quantum Zeno effect in strong coupling region. Finally, we use the Monte Carlo wave function method to numerically investigate dynamics of the quantum discord and entanglement between two atoms which are coupled to excited field in continuous dephasing process. We find the numerical simulations are similar to the analytic solution case and the values of stationary state quantum discord and entanglement is different from the case of m = 0. These results may have some potential applications in quantum information and quantum computation. Acknowledgments This project was supported by the National Natural Science Foundation of China (Grant No. 7474). References. Wheeler, J.A., Zurek, W.H.: Quantum Theory of Measurement. Princeton University Press, Princeton (983). Braginsky, V.B., Khalili, F. Ya.: Quantum Measurement. Cambridge University Press, Cambridge (99) 3. Onofrio, R., Viola, L.: Lindblad approach to nonlinear Jaynes-Cummings dynamics of a trapped ion. Phys. Rev. A. 56, (997) 4. Brun, T.A.: Continuous measurements, quantum trajectories, and decoherent histories. Phys. Rev. A. 6(4), 0407 (000)

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