Abstract

In this paper, a scheme for the generation of long-living entanglement between two distant Λ-type three-level atoms separately trapped in two dissipative cavities is proposed. In this scheme, two dissipative cavities are coupled to their own non-Markovian environments and two three-level atoms are driven by the classical fields. The entangled state between the two atoms is produced by performing Bell state measurement (BSM) on photons leaving the dissipative cavities. Using the time-dependent Schördinger equation, we obtain the analytical results for the evolution of the entanglement. It is revealed that, by manipulating the detunings of classical field, the long-living stationary entanglement between two atoms can be generated in the presence of dissipation.

© 2017 Optical Society of America

1. Introduction

Quantum entanglement, as the most important resource for quantum science and technology, draws a great deal of attention in various domains [1], such as quantum teleportation [2], quantum dense coding [3], quantum cryptography [4], and quantum computation [5]. Therefore, many schemes have been proposed to generate entangled states, such as trapped ions [6, 7], quantum electrodynamics [8,9], and photon pairs [10,11].

In order to complete a quantum operation, the long-living entanglement is needed. In real physical systems, however, quantum entanglement is fragile and very easy to be destroyed due to the interaction between quantum system and environments [12–14]. Therefore, many efforts have been devoted to the dynamical evolution of entanglement in Markovian environments [15–18]. In contrast, non-Markovian dynamics shows more interesting phenomena because of the memory effect, and has been used in various quantum operations [19–23]. Up to now, extensive researches on the entangled states for two-level atoms in dissipative environments have been done [24–27]. For example, in [27], Nourmandipour et al. investigate the entanglement swapping between two two-level atoms. Their results show that the stationary entanglement between two two-level atoms can be generated in the presence of dissipation.

Compared with the two-dimensional entanglement, high-dimensional entangled states are more competitive due to the fact that three-level quantum systems provide more secure quantum key distributions than those based on two-level systems [28–30]. Therefore, extensive researches have been devoted to the generation of three-dimensional entanglement [31–35]. For example, in [34], the generation of three-dimensional entanglement of two distant atoms in Markovian environments is proposed. In practice, the dissipation of cavities is unavoidable, and generating three-dimensional entangled states in non-Markovian environments is valuable and worth studying.

In this paper, we propose a scheme for producing the entanglement between two atoms separately trapped in two dissipative cavities. We first investigate the dynamical evolution of a three-level atom in non-Markovian environments by using the time-dependent Schördinger equation. Then, we generate the entanglement between two atoms by performing Bell state measurement on photons leaving the cavities. We use negativity to quantify the amount of entanglement [36] and discuss the effect of detunings and initial atomic states on the evolution of entanglement. The rest of the paper is organized as follows: In Sec. II, we introduce the model of the atom-field coupling system, and the dynamical evolution of entanglement between the atom and the cavity field is presented in Sec. III. In Sec. IV, we produce the entangled state between two atoms by performing Bell state measurement and discuss the effect of detunings and initial atomic states on the evolution of entanglement. The conclusions are drawn in Sec. V.

2. THE MODEL

We consider a system formed by two separate dissipative cavities, each of which contains a Λ-type three-level atom with ground state (|g〉), lower and upper excited states (|f〉, |e〉) (see Fig. 1). The quantum states |gi〉, |fi〉, and |ei〉 (i = 1, 2) have the energies of ωgi, ωfi, and ωei, respectively (ħ = 1). We assume both two dissipative cavities have high quality factors. In the ith cavity, the transition |gi〉 ↔ |ei〉 is coupled to a single-mode cavity field with the coupling constant gi, while the transition |fi〉 ↔ |ei〉 is driven through a classical field with the coupling constant Ωi. Assuming that the cavity field interacts with a reservoir consisting of a set of continuous harmonic oscillators, the Hamiltonian describing the field-reservoir is given by

Hci=ωciaiai+0Bi(η)Bi(η)dη+0Gi(η)[aiBi(η)+H.c.]dη,
where ωci is the frequency of the cavity field, Gi (η) is the coupling strength between the cavity field and the reservoir, which is a function of frequency η. Bi(η) (and Bi (η)) is the creation (and annihilation) operator of the reservoir, which obeys the commutation relation of [Bi(η),Bj(η)]=δijδ(ηη). The model of the field-reservoir shows that the dissipative cavity has a Lorentzian spectral density implying the nonperfect reflectivity of the cavity mirrors. Supposing that the reservoir has a narrow bandwidth, we can extend integrals over η from 0 to −∞ and take Gi (η) as a constant. Thus, by introducing the dressed operator Ai(ω)=αi(ω)ai+βi(ω,η)Bi(η)dη, one is able to diagonalize the Hamiltonian (1) as [37]
Hci=ωAi(ω)Ai(ω)dω.
The annihilation operator ai is given by
ai=αi*(ω)Ai(ω)dω,
with
αi(ω)=κi/πωωci+iκi,
where κi is the decay rate of the ith cavity. Consequently, the total Hamiltonian of the atom-field system is
Hi=ωAi(ω)Ai(ω)dω+ωei|eiei|+ωfi|fifi|+ωgi|gigi|+gi[αi*(ω)Ai(ω)|eigi|+H.c.]dω+Ωi[|eifi|eiωlit+H.c.],
where ωli is the frequency of the classical field in the ith cavity. Without loss of generality, we assume the atoms and the cavities have the same parameters, i.e., ωe1 = ωe2ωe, ωf1 = ωf2ωf, ωg1 = ωg2ωg, ωc1 = ωc2ωc, κ1 = κ2κ, ωl1 = ωl2ωl, g1 = g2g, and Ω1 = Ω2 ≡ Ω. In the interaction picture, the interaction Hamiltonian is given by
HIi=g[α*(ω)A(ω)|eigi|ei(ωeωgω)t+H.c.]dω+Ω[|eifi|eiΔlt+H.c.],
where Δl = ωl − (ωeωf) is the detuning of the classical field. Assuming the atom is initially in the coherent superposition of the quantum states |fi〉 and |gi〉, and the cavity field is in the vacuum state |0〉, the initial wave function of the subsystem is given by
|ψ(0)i=[cos(θi/2)|fi+sin(θi/2)eiφi|gi]|0i,
where θi ∈ [0, π], φi ∈ [0, 2π] and |0〉i represents for the vacuum state of the i environments. |1ωi = A(ω)|0〉i represents that there is one photon at frequency ω in the i environments. With at most only one excitation, the wave function of the subsystem at any time t can be written as
|ψ(t)i=[Ei(t)|ei+Fi(t)|fi+Gi(t)|gi]|0i+Ui(t,ω)|gi|1ωidω,
where Ei (t), Fi (t), Gi (t), and Ui (t) are the probability amplitudes which should be determined. Using the Schördinger equation, we obtain
E˙i(t)=igα*(ω)ei(ωeωgω)tUi(ω,t)dωiΩeiΔltFi(t)
F˙i(t)=iΩeiΔltEi(t)
G˙i(t)=0
U˙i(t)=igα(ω)ei(ωeωgω)tEi(t)
The differential equations can be solved as Gi (t) = Gi (0) = sin(θi/2)ei. Performing time integration of Eq. (10) and Eq. (12) and substituting the results into Eq. (9), we obtain
E˙i(t)=0tf(tt1)Ei(t1)dt1Ω20teiΔl(tt2)Ei(t2)dt2iΩFi(0)eiΔlt,
where the correlation function f(tt1)=J(ω)ei(ωeωgω)(tt1)dω. J(ω) is the spectral densities, which is chosen as Lorentzian function
J(ω)=g2|α(ω)|2=1πg2κ(ωωc)2+κ2.
In Eq. (14), τg = g−1 is related to the relaxation time of the system and τκ = κ−1 is the correlation time of the reservoir. When the correlation time of the reservoir is greater than the relaxation time (τκτg), the system is coupled to non-Markovian environments. Conversely, when the relaxation time is greater than the correlation time of the reservoir (τgτκ), the system is coupled to Markovian environments. Substituting the spectral densities into the correlation function, the correlation function can be written as
f(tt1)=g2e(κ+iΔ)(tt1),
where Δ = ωcωe + ωg is the detuning of the cavity field. The integro-differential equation (13) can be written as
E˙i(t)=g20te(κ+iΔ)(tt2)Ei(t1)dt1Ω20teiΔl(tt1)Ei(t2)dt2iΩFi(0)eiΔlt.
In Eq. (16), the first term is the interaction between the atom and the cavity field, which leads to the dissipation of the quantum system; the remaining terms are the interaction between the atom and the classical field. With the help of Laplace transform, we solve the integro-differential equation Eq. (16) exactly. The result is expressed by
Ei(t)=Fi(0)k=1k=3ckeskt,
where sk is the kth root of the cubic equation s3 + [i(Δ + Δl) + κ]s2 + [g2 + Ω2 + iΔl (iΔ + κ)]s + Ω2(iΔ + κ) + ig2Δl = 0, and c1 = −iΩ(s1 + iΔ + κ)/((s1s2)(s1s3)), c2 = −iΩ(s2 + iΔ + κ)/((s2s1)(s2s3)), and c3 = −iΩ(s3 + iΔ + κ)/((s3s2)(s3s1)).

 figure: Fig. 1

Fig. 1 Schematic representation of the setup. BSM is performing Bell state measurement on photons leaving the cavities.

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3. THE ENTANGLEMENT BETWEEN THE ATOM AND THE CAVITY FIELD

Let us introduce the linear entropy to quantify the entanglement between the atom and the cavity field, which is defined as

SA(θ,φ,t)=1Tr(ρA2),
where ρA is the atomic reduced density matrix of each subsystem. The range of the linear entropy is between 0 for pure state and 1 − 1/d for completely mixed state, where d is the dimension of the density matrix (here d = 3). Using Eq. (8), we obtain the atomic reduced density matrix ρA as follows
ρA=(|Ei(t)|2Ei(t)Fi*(t)Ei(t)Gi*(t)Fi(t)Ei*(t)|Fi(t)|2Fi(t)Gi*(t)Gi(t)Ei*(t)Gi(t)Fi*(t)1|Ei(t)|2|Fi(t)|2)
In this paper, we calculate the average linear entropy with respect to all possible input states on the surface of the Bloch sphere as [38]
SAav(t)=14πSA(θ,φ,t)sin(θ)dθdφ.

Figure 2 illustrates the evolution of entanglement between the atom and the cavity field over the scaled time τ = κt in non-Markovian environments. In Fig. 2, the average linear entropy exhibits an oscillatory behaviour for the memory effect of non-Markovian environments. In the absence of the detuning, the average linear entropy decays rapidly. In the presence of the detunings, the average linear entropy first increases to a maximum and then gradually decreases. Figure 3(a) shows the evolution of the populations of the states |e〉, |f〉, and |g〉 in Eq. (8) (Pe, Pf, and Pg) for the initial state |f〉 (θ = 0). In Fig. 3(a), the populations of the states |f〉 and |e〉 decrease from one to zero and the population of the state |g〉 increases from zero to one. That is because the state |f〉 is transferred into the state |g〉 through the transition path |f〉 → |e〉 → |g〉 due to the dissipation of the cavities. From Eq. (8), we know that the atom and the cavity field are in the entangled state, when 0 < Pg (t) < 1. In other words, the atom-field system will disentangle, when Pg (t) = 1. Figure 3(b) shows the evolution of the populations of state |g〉 over the scaled time τ = κt for different detunings. It shows that the population of the state |g〉 increases from zero to one more slowly in the presence of detunings, i.e., the presence of detunings can preserve the entanglement between the atom and the cavity field. It is because that the presence of the detuning Δ (Δl) decreases the transition rate between the states |e〉 and |g〉 (|f〉 and |e〉). As a result, the decay of the entanglement between the atom and the cavity field becomes slow in the presence of detunings. In addition, the amplitude of the oscillations is associated with the intensity of the memory effect of non-Markovian environments. In Fig. 2, the linear entropy shows more intensive oscillations in the absence of the detuning. In other words, the detunings suppress the memory effect of non-Markovian environments. Hence, the detunings not only make the evolution of the system slow, but also suppress the the memory effect in non-Markovian environments.

 figure: Fig. 2

Fig. 2 The evolution of the average linear entropy as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = 0, Δl = 0 (orange curve), Δ = 15κ, Δl = 0 (green curve), Δ = 0, Δl = −15κ (red curve), and Δ = 15κ, Δl = −15κ (blue curve). Other parameters: g = Ω = 10κ.

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 figure: Fig. 3

Fig. 3 (a) The evolution of the populations of the states |e〉, |f〉, and |g〉 as a function of the scaled time τ = κt for the initial state |f〉 : the population of the state |e〉 (blue curve), the population of the state |f〉 (red curve), and the population of the state |e〉 (green curve). (b) The evolution of the populations of the state |g〉 as a function of the scaled time τ = κt for different detunings: Δ = 0, Δl = 0 (orange curve), Δ = 15κ, Δl = 0 (green curve), Δ = 0, Δl = −15κ (red curve), and Δ = 15κ, Δl = −15κ (blue curve). Other common parameters: Δ = Δl = 0, g = Ω = 10κ, and θ = φ = 0.

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In Fig. 4, we plot the linear entropy of the atom-field at the scaled time τ = 15κt, as a function of the detunings Δ and Δl for initial atomic state |f〉. Figure 4 shows that when the detuning Δ = 0 (Δl = 0), the sign of the detuning Δl (Δ) has no effect on the decay of the entanglement. However, the decay of the entanglement can be suppressed, when the detunings satisfy the condition Δ · Δl < 0. That means the decay of the entanglement can be suppressed greatly by choosing the sign of the detunings Δ and Δl.

 figure: Fig. 4

Fig. 4 The linear entropy of the atom-field at the scaled time τ = 15κt, as a function of detunings Δ and Δl for initial atomic state |f〉. Other parameters: g = Ω = 10κ.

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In order to investigate the differences between the Markovian dynamics and the non-Markovian dynamics, we plot the evolution of the linear entropy between the atom and the cavity field over the scaled time τ = gt in Markovian and non-Markovian environments in Fig. 5. It shows that the linear entropy has an obvious oscillation and evolves more slowly in non-Markovian environments for the memory effect. In addition, the presence of detunings can preserve the entanglement between the atom and the cavity field both in the Markovian and non-Markovian environments.

 figure: Fig. 5

Fig. 5 The evolution of the linear entropy between the atom and the cavity field over the scaled time τ = gt for different detunings in Markovian and non-Markovian environments: Δ = Δl = 0 (orange curve), Δ = 1.5g, Δl = 0 (green curve), Δ = 0, Δl = −1.5g (red curve), and Δ = 1.5g, Δl = −1.5g (blue curve). The dashed and solid lines denote Markovian and non-Markovian environments, respectively. Other parameters: θ = φ = 0.

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4. THE ENTANGLEMENT BETWEEN TWO ATOMS

Due to the fact that the two subsystems are independent, the wave function of the total system can be written as

|ψ(t)=|ψ1(t)|ψ2(t).
However, the atom and the cavity field in each subsystem are in the entangled state. This allows us to establish the entanglement between the two atoms by performing Bell state measurement on photons leaving the cavities. We consider the Bell state
|Ψ+=12(|01|12|11|02),
where |1=Θ(ω)|1ωidω, Θ(ω) is the pulse shape associated with the coming photon. Then, acting the projection operator PF = |Ψ+〉〈Ψ+| on the wave function |ψ(t)〉 (after normalization), we obtain
|ψAA(t)=Ψ+|ψ(t)=1N(t)[X12(t)|e,gX21(t)|g,e+Y12(t)|f,gY21(t)|g,f+(Z12(t)Z21(t)|g,g)]
where
N(t)=|X12(t)|2+|X21(t)|2+|Y12(t)|2+|Y21(t)|2+|Z12(t)Z21(t)|2.
Here we have defined
Xij(t)=Ei(t)Uj(ω,t)Θ*(ω)dω,
Yij(t)=Fi(t)Uj(ω,t)Θ*(ω)dω,
Zij(t)=Gi(t)Uj(ω,t)Θ*(ω)dω.
In order to quantify the amount of entanglement between two atoms, we introduce the negativity [34], which is defined as
N(ρ(t))=ρTA12,
where ρTA is the partial transpose of ρ and XtrXX is the trace norm. Similarly, we calculate the average negativity with respect to all possible input pure separable states as
Nav(ρ(t))=116π2N(ρ(t))k=12sin(θk)dθkdφk.

Figure 6(a) illustrates the evolution of the average negativity, as a function of the scaled time τ = κt in non-Markovian environments. In Fig. 6(a), the average negativity exhibits an oscillatory decay behaviour in the absence and presence of detuning for the interaction between cavities and environments. In the presence of the detunings Δ and Δl, the decay of the entanglement between two atoms becomes slow. It is because that the entanglement between the two atoms depends on the entanglement between the atom and cavity field, i.e., the disentanglement between the atom and the cavity field will lead to the disentanglement between the two atoms. From the results in section 3, we know that the presence of detunings can preserve the entanglement between the atom and the cavity field, namely, the detunings can preserve the entanglement between the two atoms. Therefore, by choosing the detunings Δ and Δl, a long-living stationary entangled state between two atoms can be created. Figure 6(b) shows the density matrix of two atoms at the scaled time τ = 1κ. Due to the dissipation of cavities, the population of the state |g, g〉 is increased and the populations of the other states are decreased.

 figure: Fig. 6

Fig. 6 (a) The evolution of the average negativity as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = Δl = 0 (orange curve), Δ = 15κ, Δl = 0 (green curve), Δ = 0, Δl = −15κ (red curve), and Δ = 15κ, Δl = −15κ (blue curve). (b) Density matrix of the two atoms at τ = 1κt: Δ1 = Δ2 = 15κ, Δl1 = Δl2 = −15κ, and θ1 = θ2 = φ1 = φ2 = 0. Other common parameters: g = Ω = 10κ.

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On the other hand, we consider the effect of the initial atomic states on the evolution of the entanglement between two atoms. In Fig. 7, we plot the evolution of the negativity, as a function of the scaled time τ = κt for different initial atomic states in (a) non-Markovian environments and (b) Markovian environments. It is revealed that, when two atoms are in the quantum state |f〉 initially, the long-living stationary entanglement between two atoms is generated in both Markovian and non-Markovian environments. Furthermore, our further calculations show that when two atoms are in the same quantum state initially, a long-living stationary entangle state between two atoms can be produced in both Markovian and non-Markovian environments.

 figure: Fig. 7

Fig. 7 The evolution of the negativity between the two atoms as a function of the scaled time τ = κt for different initial atomic states in (a) non-Markovian environments and (b) Markovian environments: θ1 = θ2 = 0, φ1 = φ2 = 0 (blue curve), θ1 = π/2, θ2 = 0, φ1 = φ2 = 0 (red curve), θ1 = π/2, θ2 = π/4, φ1 = φ2 = 0 (green curve), and θ1 = π/2, θ2 = π/4, φ1 = π, φ2 = 0 (orange curve). Other parameters: (a) Δ = 15κ, Δl = −15κ and g = Ω = 10κ. (b) Δ = 0.15κ, Δl = −0.15κ and g = Ω = 0.1κ.

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5. Conclusion

In summary, we have investigated the system formed by two independent dissipative cavities, each of which contains a Λ-type three-level atom. We solve the time-dependent Schördinger equation of the subsystem and obtain the analytical results for the dynamical evolution of the atom-field system in non-Markovian environments. The results show that the atom and the cavity field are in the entangled state in non-Markovian environments and the decay of the entanglement is suppressed in the presence of detunings. We establish the entanglement between two atoms by performing Bell state measurement on photons leaving the cavities. It is revealed that, the presence of the detunings Δ and Δl can suppress the decay of the entanglement. By choosing the detunings and the initial atomic states, a long-living stationary entangled state between two distant atoms can be generated. Our results are useful to perform long-distance quantum communication, especially when long-living stationary entanglement is needed and the effect of environments cannot be neglected.

In the end, we briefly address the feasibility of experimental realization. In our proposed scheme, the two atoms are trapped in two distant cavities, respectively. That can be implemented experimentally by trapping atoms in cavity QED system [39,40]. The leaking photons from the two cavities are mixed on a beam splitter, and two detectors D1 and D2 are set to the both output ports of it. The projection on the states is realized, when one of the two detectors is clicked. The Bell state |Ψ±=12(|01|12±|11|02) can be distinguished by the detectors D1 and D2. The detection scheme can be implemented in linear optical system experimentally [41].

Funding

National Natural Science Foundation of China (NSFC) (11474077, and 11675046), Program for Innovation Research of Science in Harbin Institute of Technology (A201411, and A201412), the Fundamental Research Funds for the Central Universities (AUGA5710056414), Natural Science Foundation of Heilongjiang Province of China. (A201303), and Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).

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32. X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004). [CrossRef]  

33. Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005). [CrossRef]  

34. S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008). [CrossRef]  

35. H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009). [CrossRef]  

36. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002). [CrossRef]  

37. A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015). [CrossRef]  

38. A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

39. G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

40. J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004). [CrossRef]   [PubMed]  

41. X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

References

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  1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
    [Crossref]
  2. C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
    [Crossref] [PubMed]
  3. K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
    [Crossref] [PubMed]
  4. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661 (1991).
    [Crossref] [PubMed]
  5. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
    [Crossref] [PubMed]
  6. Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
    [Crossref]
  7. C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
    [Crossref] [PubMed]
  8. S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
    [Crossref] [PubMed]
  9. A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
    [Crossref] [PubMed]
  10. A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
    [Crossref]
  11. W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
    [Crossref] [PubMed]
  12. T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
    [Crossref]
  13. T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
    [Crossref] [PubMed]
  14. M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
    [Crossref]
  15. T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
    [Crossref] [PubMed]
  16. F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
    [Crossref] [PubMed]
  17. F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
    [Crossref]
  18. J. Song, Z. J. Zhang, Y. Xia, X. D. Sun, and Y. Y. Jiang, “Fast coherent manipulation of quantum states in open systems,” Opt. Express 24(19), 21674–21683 (2016).
    [Crossref] [PubMed]
  19. J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
    [Crossref]
  20. J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
    [Crossref] [PubMed]
  21. R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
    [Crossref] [PubMed]
  22. A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
    [Crossref]
  23. F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
    [Crossref]
  24. S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
    [Crossref]
  25. M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
    [Crossref]
  26. A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
    [Crossref]
  27. A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
    [Crossref]
  28. D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
    [Crossref] [PubMed]
  29. M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
    [Crossref]
  30. D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
    [Crossref] [PubMed]
  31. X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
    [Crossref]
  32. X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
    [Crossref]
  33. Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
    [Crossref]
  34. S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
    [Crossref]
  35. H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
    [Crossref]
  36. G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
    [Crossref]
  37. A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
    [Crossref]
  38. A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).
  39. G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).
  40. J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
    [Crossref] [PubMed]
  41. X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

2016 (5)

J. Song, Z. J. Zhang, Y. Xia, X. D. Sun, and Y. Y. Jiang, “Fast coherent manipulation of quantum states in open systems,” Opt. Express 24(19), 21674–21683 (2016).
[Crossref] [PubMed]

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

2015 (3)

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
[Crossref]

2014 (1)

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

2012 (1)

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

2010 (2)

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

2009 (3)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
[Crossref]

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

2008 (1)

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

2006 (4)

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
[Crossref]

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref] [PubMed]

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

2005 (1)

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

2004 (4)

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
[Crossref]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
[Crossref] [PubMed]

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

2003 (3)

T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

2002 (2)

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
[Crossref] [PubMed]

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

2001 (3)

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
[Crossref] [PubMed]

2000 (3)

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
[Crossref] [PubMed]

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

1998 (1)

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

1996 (1)

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

1993 (1)

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

1991 (1)

K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661 (1991).
[Crossref] [PubMed]

1981 (1)

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Aspect, A.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Benatti, F.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

Bennett, C. H.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Björk, G.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Blais, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Blanter, Y. M.

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

Bourennane, M.

M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
[Crossref]

Brassard, G.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Briegel, H. J.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
[Crossref] [PubMed]

Brukner, C.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Bruss, D.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
[Crossref] [PubMed]

Çakir, Ö.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Canning, J.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Cao, J.

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

Cerrillo, J.

J. Cerrillo and J. Cao, “Non-Markovian dynamical maps: numerical processing of open quantum trajectories,” Phys. Rev. Lett. 112(11), 110401 (2014).
[Crossref] [PubMed]

Correia, R. R.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Crépeau, C.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Davidovich, L.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Dukalski, M.

M. Dukalski and Y. M. Blanter, “Periodic revival of entanglement of two strongly driven qubits in a dissipative cavity,” Phys. Rev. A 82(5), 052330 (2010);
[Crossref]

Dung, H.T.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Eberly, J. H.

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
[Crossref]

Ekert, K.

K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67(6), 661 (1991).
[Crossref] [PubMed]

Estrada, A. F.

A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
[Crossref]

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

Fischer, R.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Floreanini, R.

F. Benatti and R. Floreanini, “Entangling oscillators through environment noise,” J. Phys. A 39(11), 2689 (2006).
[Crossref]

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

Frunzio, L.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
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Galve, F.

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

Gilboa, D.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
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Gnacinski, P.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
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Grangier, P.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

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S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
[Crossref] [PubMed]

Guthohriein, G. R.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Hayasaka, K.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
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Horodecki, R.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865 (2009).
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A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

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Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
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Jennewein, T.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Jiang, Y. Y.

Johnson, J.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Jozsa, R.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Kaneko, T.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
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M. Bourennane, A. Karlsson, and G. Björk, “Quantum key distribution using multilevel encoding,” Phys. Rev. A 64(1), 012306 (2001).
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Kaszlikowski, D.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

Keller, M.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Kielpinski, D.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Kiesel, N.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Kim, J.

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
[Crossref]

King, B. E.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Knöll, L.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Kofler, J.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Krischek, R.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Kwiat, P. G.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Lange, W.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Langer, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Leibfried, D.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Li, G.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Ma, X. S.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Macchiavello, C.

D. Bruss and C. Macchiavello, “Optimal eavesdropping in cryptography with three-dimensional quantum states,” Phys. Rev. Lett. 88(12), 127901 (2002).
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Majer, J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Mancini, S.

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

Mathis, W.

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
[Crossref]

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

Mattle, K.

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Meyer, V.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Michelberger, P.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Miklaszewski, W.

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

Milman, P.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Monroe, C.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Myatt, C. J.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Nourmandipour, A.

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

Oh, S.

S. Oh and J. Kim, “Entanglement between qubits induced by a common environment with a gap,” Phys. Rev. B 73(6), 062306 (2006).
[Crossref]

Pachón, L. A.

F. Galve, L. A. Pachón, and D. Zueco, “Ultrafast optimal sideband cooling under non-Markovian evolution,” Phys. Rev. Lett. 116(18), 183602 (2016).
[Crossref]

A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
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A. F. Estrada, L. A. Pachón, and L. A. Pachón, “Quantum limit for driven linear non-Markovian open-quantum-systems,” New J. Phys. 17(3), 033038 (2015).
[Crossref]

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

Pahlke, K.

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

Peres, A.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Piani, M.

F. Benatti, R. Floreanini, and M. Piani, “Environment induced entanglement in Markovian dissipative dynamics,” Phys. Rev. Lett. 91(7), 070402 (2003).
[Crossref] [PubMed]

Prado, S. D.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Pru, J. K.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Rafiee, M.

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

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R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett. 86(22), 5188 (2001).
[Crossref] [PubMed]

Ribeiro-Teixeira, A. C.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Roger, G.

A. Aspect, P. Grangier, and G. Roger, “Experimental tests of realistic local theories via Bell’s theorem,” Phys. Rev. Lett. 47(7), 460 (1981).
[Crossref]

Sackett, C. A.

C. A. Sackett, D. Kielpinski, B. E. King, C. Langer, V. Meyer, C. J. Myatt, and C. Monroe, “Experimental entanglement of four particles,” Nature 404(6775), 256–259 (2000).
[Crossref] [PubMed]

Santos, M. F.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Schoelkopf, R. J.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Schuster, D. I.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Silberberg, Y.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Song, J.

Sun, X. D.

Tan, H.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Tavassoly, M. K.

A. Nourmandipour, M. K. Tavassoly, and S. Mancini, “The entangling power of a glocal dissipative map,” Quantum Inf. Comput. 16(11), 0969–0981 (2016).

A. Nourmandipour and M. K. Tavassoly, “Entanglement swapping between dissipative systems,” Phys. Rev. A 94(2), 022339 (2016).
[Crossref]

A. Nourmandipour, M. K. Tavassoly, and M. Rafiee, “Dynamics and protection of entanglement in n-qubit systems within Markovian and non-Markovian environments,” Phys. Rev. A 93(2), 022327 (2016).
[Crossref]

A. Nourmandipour and M. K. Tavassoly, “Dynamics and protecting of entanglement in two-level systems interacting with a dissipative cavity: the Gardiner-Collett approach,” J. Phys. B: At. Mol. Opt. Phys. 48(16), 165502 (2015).
[Crossref]

Tilly, J. L.

J. Johnson, J. Canning, T. Kaneko, J. K. Pru, and J. L. Tilly, “Germline stem cells and follicular renewal in the postnatal mammalian ovary,” Nature 428(6979), 145–150 (2004).
[Crossref] [PubMed]

Tóth, G.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Triana, J. F.

J. F. Triana, A. F. Estrada, and L. A. Pachón, “Bringing entanglement to the high temperature limit,” Phys. Rev. Lett. 105(18), 180501 (2010).
[Crossref]

Turchette, Q. A.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Ursin, R.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Vidal, G.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

Vidal, I.

R. Fischer, I. Vidal, D. Gilboa, R. R. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

Wallraff, A.

A. Wallraff, D. I. Schuster, A. Blais, L. Frunzio, R. S. Huang, J. Majer, and R. J. Schoelkopf, “Strong coupling of a single photon to a superconducting qubit using circuit quantum electrodynamics,” Nature 431(7005), 162–167 (2004).
[Crossref] [PubMed]

Walther, H.

G. R. Guthohriein, M. Keller, K. Hayasaka, W. Lange, and H. Walther, “A single ion as a nanoscopic probe of an optical field,” Nature 414(6859), 256–259 (2001).

Weinfurter, H.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Welsch, D. G.

Ö. Çakir, H.T. Dung, L. Knöll, and D. G. Welsch, “Generation of long-living entanglement between two separate three-level atoms,” Phys. Rev. A 71(3), 032326 (2005).
[Crossref]

Werner, R. F.

G. Vidal and R. F. Werner, “Computable measure of entanglement,” Phys. Rev. A 65(3), 032314 (2002).
[Crossref]

Wieczorek, W.

W. Wieczorek, R. Krischek, N. Kiesel, P. Michelberger, G. Tóth, and H. Weinfurter, “Experimental entanglement of a six-photon symmetric Dicke state,” Phys. Rev. Lett. 103(2), 020504 (2009).
[Crossref] [PubMed]

Wineland, D. J.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Wood, C. S.

Q. A. Turchette, C. S. Wood, B. E. King, C. J. Myatt, D. Leibfried, W. M. Itano, and D. J. Wineland, “Deterministic entanglement of two trapped ions,” Phys. Rev. Lett. 81(17), 3631 (1998).
[Crossref]

Wootters, W. K.

C. H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, and W. K. Wootters, “Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels,” Phys. Rev. Lett. 70(13), 1895 (1993).
[Crossref] [PubMed]

Xia, Y.

Xia H, H.

H. Tan, H. Xia H, and G. Li, “Interference-induced enhancement of field entanglement from an intracavity three-level V-type atom,” Phys. Rev. A 79(6), 063805 (2009).
[Crossref]

Ye, S. Y.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

Yu, T.

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97(14), 140403 (2006).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Finite-time disentanglement via spontaneous emission,” Phys. Rev. Lett. 93(14), 140404 (2004).
[Crossref] [PubMed]

T. Yu and J. H. Eberly, “Qubit disentanglement and decoherence via dephasing,” Phys. Rev. B 68(16), 165322 (2003).
[Crossref]

Zagury, N.

M. F. Santos, P. Milman, L. Davidovich, and N. Zagury, “Direct measurement of finite-time disentanglement induced by a reservoir,” Phys. Rev. B 73(4), 040305 (2006).
[Crossref]

Zeilinger, A.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

D. Kaszlikowski, P. Gnaciński, M. Żukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85(21), 4418 (2000).
[Crossref] [PubMed]

K. Mattle, H. Weinfurter, P. G. Kwiat, and A. Zeilinger, “Dense coding in experimental quantum communication,” Phys. Rev. Lett. 76(25), 4656 (1996).
[Crossref] [PubMed]

Zhang, Z. J.

Zheng, S. B.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

S. B. Zheng and G. C. Guo, “Efficient scheme for two-atom entanglement and quantum information processing in cavity QED,” Phys. Rev. Lett. 85(11), 2392 (2000).
[Crossref] [PubMed]

Zhong, Z. R.

S. Y. Ye, Z. R. Zhong, and S. B. Zheng, “Deterministic generation of three-dimensional entanglement for two atoms separately trapped in two optical cavities,” Phys. Rev. A 77(1), 014303 (2008).
[Crossref]

Zotter, S.

X. S. Ma, S. Zotter, J. Kofler, R. Ursin, T. Jennewein, Č. Brukner, and A. Zeilinger, “Experimental delayed-choice entanglement swapping,” Nature 8(6), 479–484 (2012).

Zou, X.

X. Zou and W. Mathis, “One-step implementation of maximally entangled states of many three-level atoms in microwave cavity QED,” Phys. Rev. A 70(3), 035802 (2004).
[Crossref]

X. Zou, K. Pahlke, and W. Mathis, “Generation of an entangled state of two three-level atoms in cavity QED,” Phys. Rev. A 67(4), 044301 (2003).
[Crossref]

Zueco, D.

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Figures (7)

Fig. 1
Fig. 1 Schematic representation of the setup. BSM is performing Bell state measurement on photons leaving the cavities.
Fig. 2
Fig. 2 The evolution of the average linear entropy as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = 0, Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). Other parameters: g = Ω = 10κ.
Fig. 3
Fig. 3 (a) The evolution of the populations of the states |e〉, |f〉, and |g〉 as a function of the scaled time τ = κt for the initial state |f〉 : the population of the state |e〉 (blue curve), the population of the state |f〉 (red curve), and the population of the state |e〉 (green curve). (b) The evolution of the populations of the state |g〉 as a function of the scaled time τ = κt for different detunings: Δ = 0, Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). Other common parameters: Δ = Δ l = 0, g = Ω = 10κ, and θ = φ = 0.
Fig. 4
Fig. 4 The linear entropy of the atom-field at the scaled time τ = 15κt, as a function of detunings Δ and Δ l for initial atomic state |f〉. Other parameters: g = Ω = 10κ.
Fig. 5
Fig. 5 The evolution of the linear entropy between the atom and the cavity field over the scaled time τ = gt for different detunings in Markovian and non-Markovian environments: Δ = Δ l = 0 (orange curve), Δ = 1.5g, Δ l = 0 (green curve), Δ = 0, Δ l = −1.5g (red curve), and Δ = 1.5g, Δ l = −1.5g (blue curve). The dashed and solid lines denote Markovian and non-Markovian environments, respectively. Other parameters: θ = φ = 0.
Fig. 6
Fig. 6 (a) The evolution of the average negativity as a function of the scaled time τ = κt for different detunings in non-Markovian environments: Δ = Δ l = 0 (orange curve), Δ = 15κ, Δ l = 0 (green curve), Δ = 0, Δ l = −15κ (red curve), and Δ = 15κ, Δ l = −15κ (blue curve). (b) Density matrix of the two atoms at τ = 1κt: Δ1 = Δ2 = 15κ, Δ l 1 = Δ l 2 = −15κ, and θ1 = θ2 = φ1 = φ2 = 0. Other common parameters: g = Ω = 10κ.
Fig. 7
Fig. 7 The evolution of the negativity between the two atoms as a function of the scaled time τ = κt for different initial atomic states in (a) non-Markovian environments and (b) Markovian environments: θ1 = θ2 = 0, φ1 = φ2 = 0 (blue curve), θ1 = π/2, θ2 = 0, φ1 = φ2 = 0 (red curve), θ1 = π/2, θ2 = π/4, φ1 = φ2 = 0 (green curve), and θ1 = π/2, θ2 = π/4, φ1 = π, φ2 = 0 (orange curve). Other parameters: (a) Δ = 15κ, Δ l = −15κ and g = Ω = 10κ. (b) Δ = 0.15κ, Δ l = −0.15κ and g = Ω = 0.1κ.

Equations (29)

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H c i = ω c i a i a i + 0 B i ( η ) B i ( η ) d η + 0 G i ( η ) [ a i B i ( η ) + H . c . ] d η ,
H c i = ω A i ( ω ) A i ( ω ) d ω .
a i = α i * ( ω ) A i ( ω ) d ω ,
α i ( ω ) = κ i / π ω ω c i + i κ i ,
H i = ω A i ( ω ) A i ( ω ) d ω + ω e i | e i e i | + ω f i | f i f i | + ω g i | g i g i | + g i [ α i * ( ω ) A i ( ω ) | e i g i | + H . c . ] d ω + Ω i [ | e i f i | e i ω l i t + H . c . ] ,
H I i = g [ α * ( ω ) A ( ω ) | e i g i | e i ( ω e ω g ω ) t + H . c . ] d ω + Ω [ | e i f i | e i Δ l t + H . c . ] ,
| ψ ( 0 ) i = [ cos ( θ i / 2 ) | f i + sin ( θ i / 2 ) e i φ i | g i ] | 0 i ,
| ψ ( t ) i = [ E i ( t ) | e i + F i ( t ) | f i + G i ( t ) | g i ] | 0 i + U i ( t , ω ) | g i | 1 ω i d ω ,
E ˙ i ( t ) = i g α * ( ω ) e i ( ω e ω g ω ) t U i ( ω , t ) d ω i Ω e i Δ l t F i ( t )
F ˙ i ( t ) = i Ω e i Δ l t E i ( t )
G ˙ i ( t ) = 0
U ˙ i ( t ) = i g α ( ω ) e i ( ω e ω g ω ) t E i ( t )
E ˙ i ( t ) = 0 t f ( t t 1 ) E i ( t 1 ) d t 1 Ω 2 0 t e i Δ l ( t t 2 ) E i ( t 2 ) d t 2 i Ω F i ( 0 ) e i Δ l t ,
J ( ω ) = g 2 | α ( ω ) | 2 = 1 π g 2 κ ( ω ω c ) 2 + κ 2 .
f ( t t 1 ) = g 2 e ( κ + i Δ ) ( t t 1 ) ,
E ˙ i ( t ) = g 2 0 t e ( κ + i Δ ) ( t t 2 ) E i ( t 1 ) d t 1 Ω 2 0 t e i Δ l ( t t 1 ) E i ( t 2 ) d t 2 i Ω F i ( 0 ) e i Δ l t .
E i ( t ) = F i ( 0 ) k = 1 k = 3 c k e s k t ,
S A ( θ , φ , t ) = 1 Tr ( ρ A 2 ) ,
ρ A = ( | E i ( t ) | 2 E i ( t ) F i * ( t ) E i ( t ) G i * ( t ) F i ( t ) E i * ( t ) | F i ( t ) | 2 F i ( t ) G i * ( t ) G i ( t ) E i * ( t ) G i ( t ) F i * ( t ) 1 | E i ( t ) | 2 | F i ( t ) | 2 )
S A av ( t ) = 1 4 π S A ( θ , φ , t ) sin ( θ ) d θ d φ .
| ψ ( t ) = | ψ 1 ( t ) | ψ 2 ( t ) .
| Ψ + = 1 2 ( | 0 1 | 1 2 | 1 1 | 0 2 ) ,
| ψ A A ( t ) = Ψ + | ψ ( t ) = 1 N ( t ) [ X 12 ( t ) | e , g X 21 ( t ) | g , e + Y 12 ( t ) | f , g Y 21 ( t ) | g , f + ( Z 12 ( t ) Z 21 ( t ) | g , g ) ]
N ( t ) = | X 12 ( t ) | 2 + | X 21 ( t ) | 2 + | Y 12 ( t ) | 2 + | Y 21 ( t ) | 2 + | Z 12 ( t ) Z 21 ( t ) | 2 .
X i j ( t ) = E i ( t ) U j ( ω , t ) Θ * ( ω ) d ω ,
Y i j ( t ) = F i ( t ) U j ( ω , t ) Θ * ( ω ) d ω ,
Z i j ( t ) = G i ( t ) U j ( ω , t ) Θ * ( ω ) d ω .
N ( ρ ( t ) ) = ρ T A 1 2 ,
N a v ( ρ ( t ) ) = 1 16 π 2 N ( ρ ( t ) ) k = 1 2 sin ( θ k ) d θ k d φ k .

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