1. Introduction
Quantum entanglement is not only a striking feature of quantum mechanics, but also a valuable physical resource in quantum information processing, such as quantum key distribution [1–3], quantum dense coding [4], quantum teleportation [5–7], quantum secret sharing [8] and quantum secure direct communication [9–12]. With the development of quantum optics and quantum information science, much effort has been devoted to prepare and manipulate entangled states in various quantum systems [13–20]. However, recent works have revealed that entanglement is not the only type of quantum correlation because it does not account for all of the non-classical properties of quantum correlation. While quantum discord (QD) first introduced by Zureks [21] is supposed to have the ability to capture all quantum correlation in a quantum system. QD provides a speedup for some certain tasks over their classical counterparts both theoretically [22, 23] and experimentally [24]. In this sense, QD is believed to be a new resource for quantum information.
On the other hand, a real quantum system will inevitably interact with the environment, which causes the real quantum system to be open to the external environment. The interaction between the system and environment will destroy the quantum coherence of the system and bring obstacles in preparing and preserving entangled states. Therefore, studying essential features of dynamical behavior of entanglement under decoherence has attracted much attention in recent years [25–38]. Among these achievements, the most striking discovery is the entanglement sudden death (ESD) [25], which describes the finite time disentanglement of two qubits coupled with two independent reservoirs. Shortly afterwards, the authors in [27] have shown that the ESD of a two-qubit system is intimately linked to sudden birth of entanglement (ESB) between two independent reservoirs. In addition, a prerequisite for efficient quantum computation is that quantum operations through quantum devices can maintain coherence over a longer period of time [39]. Therefore, it is of great significance to study the dynamical properties of open systems and to suppress quantum decoherence [40, 41]. In recent years, the influence of initial correlation on the open system dynamics has been intensively studied [42–48]. In particular, the authors in [49, 50] have discovered that the initial system-reservoir correlations can initially increase the entanglement and QD between a two-qubit system. We note in these studies only the dynamics of entanglement and QD of two qubits interacting with a common structured Markovian and non-Markovian reservoirs are considered, and the case of independent structured reservoirs is scarcely studied. Motivated by this, in this paper we focus on the dynamics of two qubits interacting with two independent structured reservoirs in presence of initial system-reservoir correlations. We set three different initial states by distinguishing initial system-reservoir correlations and analyze the effects of system-reservoir correlations on the dynamical properties of the two qubits system.
The paper is organized as follows. In section 2 , we give the research model and briefly introduce the solution method of master equation. In section 3 , we first review the theoretical knowledge of the quantum correlation measurement, i.e. the concurrence and QD, and then give our numerical results and discussion in detail. Finally, we summarize the paper in section 4 .
2. Model
When a two-level atom interacts with a electromagnetic field, the total Hamiltonian of the ‘qubit+reservior’ in the rotating-wave approximation is given by $ \begin{eqnarray}H={\omega }_{0}{\sigma }_{+}{\sigma }_{-}+\sum _{k}{\omega }_{k}{a}_{k}^{\dagger }{a}_{k}+\left(\sum _{k}{g}_{k}{\sigma }_{+}{a}_{k}+{\rm{h}}{\rm{.}}{\rm{c}}\left.{\rm{.}}\Space{0ex}{3.8ex}{0ex}\right)\right.,\end{eqnarray}$ where ω0 is the transition frequency and Σ± are the rasing and lowering operators of the atom. ak and ${a}_{k}^{\dagger }$ are the annihilation and creation operators of the field mode k with frequency ωk. gk is the qubit-reservoir coupling constant. If we assume the atom interacts resonantly with a Lorentzian structured reservoir, i.e. the electromagnetic field inside a lossy resonator, with the help of pseudomode approach, then we can obtain the dynamics of the atom by solving the pseudomode master equation $ \begin{eqnarray}\frac{{\rm{d}}\rho }{{\rm{d}}t}=-{\rm{i}}[V,\rho ]-\frac{{\rm{\Gamma }}}{2}\left[{a}^{\dagger }a\rho -2a\rho {a}^{\dagger }+\rho {a}^{\dagger }a\right],\end{eqnarray}$ where ρ is the density matrix of the pseudomode and the atom, and $V={\rm{\Omega }}({\sigma }_{+}a+{\sigma }_{-}{a}^{\dagger })$ . a and a† are the annihilation and creation operators of the pseudomode, ω and Γ describe the coupling strength between the atom and the pseudomode and the decay rate of the pseudomode respectively. They both depend on the Lorentzian spectral distribution $J(\omega )=\tfrac{{{\rm{\Omega }}}^{2}}{2\pi }\tfrac{{\rm{\Gamma }}}{{\left(\omega -{\omega }_{0}\right)}^{2}+{\left(\tfrac{{\rm{\Gamma }}}{2}\right)}^{2}}$ .
In this paper, we consider the dynamics of two two-level atoms interacting with two coupled Lorentzian reservoirs. Based on the above discussion, this is equivalent to the fact that the two qubits interact with two dissipative coupled cavities respectively. For this case, the dynamics of the two qubits can be treated in the following pseudomode master equation $ \begin{eqnarray}\begin{array}{rcl}\frac{{\rm{d}}\rho }{{\rm{d}}t} & = & -{\rm{i}}[V,\rho ]-\frac{{{\rm{\Gamma }}}_{1}}{2}\left[{a}_{1}^{\dagger }{a}_{1}\rho -2{a}_{1}\rho {a}_{1}^{\dagger }+\rho {a}_{1}^{\dagger }{a}_{1}\right]\\ & & -\frac{{{\rm{\Gamma }}}_{2}}{2}\left[{a}_{2}^{\dagger }{a}_{2}\rho -2{a}_{2}\rho {a}_{2}^{\dagger }+\rho {a}_{2}^{\dagger }{a}_{2}\right],\end{array}\end{eqnarray}$ with $ \begin{eqnarray}V=\left({{\rm{\Omega }}}_{1}{\sigma }_{-}^{A}{a}_{1}^{\dagger }+{{\rm{\Omega }}}_{2}{\sigma }_{-}^{B}{a}_{2}^{\dagger }+{\rm{h}}{\rm{.}}{\rm{c}}.\right)+\nu \left({a}_{1}^{\dagger }{a}_{2}+{a}_{1}{a}_{2}^{\dagger }\right),\end{eqnarray}$ where ν is inter-cavity hopping rate of photons between the two cavities. For the sake of simplicity, we let ω1 = ω2 = ω and Γ1 = Γ2 = Γ. According to [49], the dynamics of the qubits is qualitatively distinguished in two regimes: the weak coupling regime Γ/ω > 2 corresponding to Markovian dynamics and the strong coupling regime Γ/ω < 2 which exhibits non-Markovian dynamics. Our aim is to investigate the effects of initial atom-cavities correlations on the dynamics of the atomic quantum correlation. Furthermore, the initial correlation between the two cavities is also taken into account. Therefore, we considered three initial states: $ \begin{eqnarray}{\rho }_{{ABE}}^{(1)}=| {\rm{\Psi }}{\rangle }_{{ABE}}\langle {\rm{\Psi }}| ,\end{eqnarray}$ with $| {\rm{\Psi }}{\rangle }_{{ABE}}=\alpha | {ge}00\rangle +\beta | {eg}00\rangle +\tfrac{\gamma }{\sqrt{2}}(| {gg}01\rangle +| {gg}10\rangle )$ , $ \begin{eqnarray}\begin{array}{rcl}{\rho }_{{ABE}}^{(2)} & = & (\alpha | {ge}\rangle +\beta | {eg}\rangle )({\alpha }^{* }\langle {ge}| \,+\,{\beta }^{* }\langle {eg}| )\otimes | 00\rangle \langle 00| \\ & & +\displaystyle \frac{| \gamma {| }^{2}}{2}| {gg}\rangle \langle {gg}| \otimes (| 01\rangle +| 10\rangle )(\langle 01| +\langle 10| ),\end{array}\end{eqnarray}$ and ${\rho }_{{ABE}}^{(3)}={\rho }_{{AB}}\otimes {\rho }_{E}$ with $| \alpha {| }^{2}+| \beta {| }^{2}+| \gamma {| }^{2}=1(\alpha ,\beta ,\gamma \,\ne 0)$ . For convenience, we choose the parameters $\alpha \,=\sin \theta \cos \varphi ,\beta =\sin \theta \sin \varphi $ , and $\gamma =\cos \theta $ ( $\theta \in [0,\pi /2]$ and $\varphi \in [0,\pi ])$ in the discussion. $| e\rangle $ and $| g\rangle $ are the excited and ground states of atoms, and $| 0\rangle $ and $| 1\rangle $ are the vacuum and single-photo states of the lossy cavity. In initial state ${\rho }_{{ABE}}^{(1)}$ , both the classical correlation and quantum correlation exist between the system and the cavities. In initial state ${\rho }_{{ABE}}^{(2)}$ , the system and the cavities only have classical correlation. While the system and the cavities are initially in the factorized state ${\rho }_{{ABE}}^{(3)}$ , which means there is no initial atom-cavity correlation. It is obvious to note that the different three initial states have the same reduced density matrixes for the atoms and cavities, i.e. $ \begin{eqnarray}{\rho }_{{AB}}=\left(\begin{array}{cccc}| \gamma {| }^{2} & 0 & 0 & 0\\ 0 & | \alpha {| }^{2} & \alpha {\beta }^{* } & 0\\ 0 & {\alpha }^{* }\beta & | \beta {| }^{2} & 0\\ 0 & 0 & 0 & 0\end{array}\right),\end{eqnarray}$ $ \begin{eqnarray}{\rho }_{E}=\left(\begin{array}{cccc}| \alpha {| }^{2}+| \beta {| }^{2} & 0 & 0 & 0\\ 0 & \tfrac{| \gamma {| }^{2}}{2} & \tfrac{| \gamma {| }^{2}}{2} & 0\\ 0 & \tfrac{| \gamma {| }^{2}}{2} & \tfrac{| \gamma {| }^{2}}{2} & 0\\ 0 & 0 & 0 & 0\end{array}\right).\end{eqnarray}$ In order to study the atomic dynamics, we must solve the master equation. For the initial states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ , we find the total system contains at most one excitation. In this case we need to solve a set of 5 × 5 differential equations involving the basis $| {gg}00\rangle ,| {ge}00\rangle ,| {eg}00\rangle ,| {gg}01\rangle ,| {gg}10\rangle $ . However, for the initial state ${\rho }_{{ABE}}^{(3)}$ , we must add another sets of differential equations since at most two excitations are contained in this case. Tracing out the pseudomode degree of freedom, the reduced density matrix of the two qubits for every initial state is obtained. As a result, we can analyze the effects of different initial atom-cavity correlations on the dynamics of atomic quantum correlation.
3. Numerical results and discussions
Concurrence introduced by Wootters is considered to be a powerful measurement for entanglement between any two-qubit system. The general expression of concurrence is given by [51] $C=\max \{0,{\lambda }_{1}-{\lambda }_{2}-{\lambda }_{3}-{\lambda }_{4}\}$ , where the quantities λi (i = 1..4) are the square non-zero roots of eigenvalues for the matrix $R={\rho }_{{AB}}{\sigma }_{y}\otimes {\sigma }_{y}{\rho }_{{AB}}^{* }{\sigma }_{y}\otimes {\sigma }_{y}$ in descending order. The concurrence attains its maximum value C = 1 for maximally entangled states and vanishes for separate states. On the other hand, for a bipartite quantum system, the QD is defined as [21] $D({\rho }_{{AB}})=I({\rho }_{{AB}})-{ \mathcal C }({\rho }_{{AB}})$ , where $I({\rho }_{{AB}})=S({\rho }_{A})+S({\rho }_{B})-S({\rho }_{{AB}})$ is the quantum mutual information and ${ \mathcal C }({\rho }_{{AB}})$ is the classical correlation between the two subsystems. $S({\rho }_{{AB}})=-\mathrm{tr}{\rho }_{{AB}}\mathrm{log}{\rho }_{{AB}}$ is the von Neumann entropy. The classical correlation is provided by ${ \mathcal C }({\rho }_{{AB}})={\max }_{\{{{\rm{\Pi }}}_{k}^{A}\}}[S({\rho }_{B})-{{\rm{\Sigma }}}_{k}{p}_{k}S({\rho }_{B| k})]$ where the maximum is taken over the set of projective measurements $\{{{\rm{\Pi }}}_{k}^{A}\}$ on subsystem A and ${\rho }_{B| k}={\mathrm{tr}}_{A}({{\rm{\Pi }}}_{k}^{A}{\rho }_{{AB}}{{\rm{\Pi }}}_{k}^{A})/{p}_{k}$ with ${p}_{k}={\mathrm{tr}}_{{AB}}({{\rm{\Pi }}}_{k}^{A}{\rho }_{{AB}}{{\rm{\Pi }}}_{k}^{A})$ . It is sufficient for us to evaluate the QD using the following set of projectors: $\{{{\rm{\Pi }}}_{k}^{A}\,=| {\psi }_{1}\rangle \langle {\psi }_{1}| ,| {\psi }_{2}\rangle \langle {\psi }_{2}| \}$ , in which $| {\psi }_{1} \rangle =\cos \theta | g \rangle +{{\rm{e}}}^{{\rm{i}}\varphi }\sin \theta | e \rangle $ and $| {\psi }_{2} \rangle =-\cos \theta | e \rangle +{{\rm{e}}}^{-{\rm{i}}\varphi }\sin \theta | g \rangle $ with $\theta \in [0,\pi ]$ and $\varphi \in [0,2\pi ]$ . We can obtain the QD via numerical optimization over the parameters θ and φ. Using concurrence and QD, in what follows we discuss the dynamics of the atomic quantum correlation considering the effects of initial atom-cavity correlations.
In figure 1 we give the plot of the dynamics of atomic entanglement and QD for the three different initial states with θ = φ = π/6. One can observe from figures 1(a) and (b) that in the Markovian regime both entanglement and QD increase first and then decay asymptotically for the initial states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ . However, the entanglement experiences ESD and the QD decays monotonously for the factorized state ${\rho }_{{ABE}}^{(3)}$ . In view of the fact that the reduced density matrixes of the two atoms and the cavities in the three initial states are the same, we can conclude that the initial atom-cavity correlations irrespective of quantum or classical correlation can not only avoid the occurrence of ESD but also enhance the atomic quantum correlation. Furthermore, the ability to enhance the entanglement and QD of ${\rho }_{{ABE}}^{(1)}$ is stronger than ${\rho }_{{ABE}}^{(2)}$ , which can be seen obviously in the evolution of QD. This can be explained that ${\rho }_{{ABE}}^{(1)}$ contains quantum correlation but only classical correlation contained in ${\rho }_{{ABE}}^{(2)}$ . However, this effect will be eliminated in the non-Markovian regime shown in figures 1(c) and (d). We find that atomic entanglement and QD exhibit the same intense oscillation with time under the initial correlation states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ . Meanwhile, the atomic entanglement undergoes a series of vanishes and revivals for the case ${\rho }_{{ABE}}^{(3)}$ , and its maximal revival value can exceed the initial value. In comparison to the entanglement, the atomic QD only experiences oscillatory damping. We know the revival phenomenon is induced by the memory effects of the non-Markovian reservoirs, which indicates that the entanglement is influenced more strongly than QD by the non-Markovian effect for the factorized state ${\rho }_{{ABE}}^{(3)}$ .
Figure 1. Atomic concurrence and quantum discord as a function of dimensionless quantity ωt for different initial states in Markovian regime (a), (b) and non-Markovian regime (c), (d). The parameters are chosen as |
Figure 2 shows the evolutions of atomic entanglement and QD when the two atoms are initially prepared in separated state, where we choose θ = π/3 and φ = 0. It is seen that both entanglement and QD can be generated for the initial states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ in the Markovian regime. Obviously, the initially created maximal entanglement and QD for ${\rho }_{{ABE}}^{(1)}$ are much larger than the case of ${\rho }_{{ABE}}^{(2)}$ . However, for the initial state ${\rho }_{{ABE}}^{(3)}$ , there is no entanglement generation and only a weak QD can be induced. Therefore, the initial atom-cavity correlations play an essential role in the generation of the atomic quantum correlation. To make a comparison, the corresponding dynamics of entanglement and QD in the non-Markovian regime are presented in figures 2(c) and (d). As expected, the atomic entanglement and QD exhibit a series of births and vanishes for the initial states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ . Certainly, the amplitude of state ${\rho }_{{ABE}}^{(1)}$ is larger than that of ${\rho }_{{ABE}}^{(2)}$ . For state ${\rho }_{{ABE}}^{(3)}$ , the most distinct behavior in the non-Markovian is that the ESD and ESB for the atoms occur in the evolution, and the QD vanishes only at discrete instants. This is because the non-Markovian effect characterized by the feedback of information from the reservoirs into the system enhances the appearance of atomic quantum correlation.
Figure 2. Atomic concurrence and quantum discord as a function of dimensionless quantity ωt for different initial states in Markovian regime (a), (b) and non-Markovian regime (c), (d). The parameters are chosen as θ = π/3, φ = 0. (a), (b) Γ = 5ω and (c), (d) Γ = 0.2ω. |
Considering the transference of entanglement from the cavities to atoms, we investigate the dynamics of atomic entanglement for the initial state $| \psi (0){\rangle }_{{ABE}}\,=| {gg}\rangle \otimes (\cos \zeta | 01\rangle +\sin \zeta | 10\rangle )$ in figure 3. It is expected that when the atoms and the cavities are initially in a factorized state, whether the entanglement between the two separated atoms can be generated depends on the initial entanglement between the two cavities. When ζ = 0, i.e. there is no entanglement initially between the two cavities, the atomic entanglement can not be created in the evolution. But the atomic entanglement can be induced when the two cavities are initially entangled, and the larger the degree of the initial entanglement in the two cavities, the larger maximal value of atomic entanglement is obtained. For example, for ζ = π/4, the whole system is initially the factorized state $| \psi (0){\rangle }_{{ABE}}=| {gg}\rangle \otimes \tfrac{1}{\sqrt{2}}(| 01\rangle +| 10\rangle )$ where the two cavities are prepared in the maximally entangled state, our result shows that the maximal value of atomic entanglement is largest for all ζ. As to the generation of atomic QD, we omit the details since the result is the same as the discussion about entanglement generation.
Figure 3. Atomic concurrence as a function of dimensionless quantity ωt for the initial state |
Now we turn to study the effects of inter-cavity hopping rate ν on the dynamics of atomic quantum correlation. Here we also take the entanglement evolution as an example, since the same conclusion can be obtained from the evolution of QD. Figure 4 shows the evolution of atomic entanglement for different values of ν in Markovian regime. One can see that in the long time evolution, increasing ν has the same impact on the atomic entanglement for the three initial states, that is, the decay of entanglement becomes slower as ν increases. Especially for state ${\rho }_{{ABE}}^{(3)}$ , the entanglement experiences rebirth after sudden death and the ESD can be avoided by choosing appropriate ν. The physical mechanism behind this can be explained as [52] by introducing the delocalized atomic and field operators, which reveals that a large hopping rate ν between the cavities can preserve the atomic entanglement by restraining the atoms from entangling with the cavities. However, in the early time evolution, increasing ν will enhance the decay of atomic entanglement for states ${\rho }_{{ABE}}^{(1)}$ and ${\rho }_{{ABE}}^{(2)}$ , and this is much more affected for state ${\rho }_{{ABE}}^{(1)}$ than state ${\rho }_{{ABE}}^{(2)}$ . For example, when $\nu \geqslant {\rm{\Omega }}$ , the atomic entanglement can not be initially increased and only initially decreased entanglement happened in the evolution of state ${\rho }_{{ABE}}^{(1)}$ . It can be understood that due to the initial atom-cavity correlation, especially quantum correlation, strengthening the hopping rate between the two cavities in turn generates some effective interaction between the atoms and the cavities, which counters the fast disentanglement. Furthermore, the above conclusion is also valid in non-Markovian regime by comparing the variation of amplitude of entanglement shown in figure 5.
Figure 4. Atomic concurrence as a function of dimensionless quantity ωt for different hopping rates ν in Markovian regime Γ = 5ω. (a) initial state |
Figure 5. Atomic concurrence as a function of dimensionless quantity ωt for different hopping rates ν in non-Markovian regime Γ = 0.2ω. (a) initial state |
4. Conclusion
In this paper, by the concepts of entanglement and QD, we have studied the dynamics of quantum correlation between two two-level atoms, each of which is coupled with one of two coupled Lorentzian reservoirs. In virtue of pseudomode approach, this model can be described as two qubits interacting with two coupled dissipative cavities. We focus on the effects of the initial atom-cavity correlations on the dynamics of atomic quantum correlation in both Markovian and non-Markovian regimes. Comparing with the case of initial factorized state, we find that initial atom-cavity correlations, including classical and quantum correlation, can not only protect the atomic entanglement from sudden death but also enhance the atomic quantum correlation. Furthermore, the initial atom-cavity correlations do better in strengthening QD, though it not valid in non-Markovian regime. In the study of the creation of atomic quantum correlation, we show that the initial atom-cavity correlations, especially the initial quantum correlation, is responsible for the generation of atomic entanglement and QD. Besides, when the atoms and the cavities are initially in the factorized state, the strong quantum correlation between the two cavities is beneficial to create the atomic quantum correlation. Finally, the effects of inter-cavity hopping rate on the evolution of atomic entanglement are also discussed. We find that for the initial correlation states, increasing the hopping rate will enhance the decay of atomic entanglement in the early time evolution, but it is opposite in the long time evolution.