On the relationship between non-Markovianity and entanglement protection

It is by now well known that entanglement can suddenly disappear, stay absent for some time and then reappear. These phenomena, respectively termed as 'entanglement sudden death' (ESD) [1] and 'entanglement sudden birth' (ESB) [2], appear under conditions that have been extensively studied. Recently, Yönaç et al [3] considered a four-qubit system and analysed entanglement for its six possible partitions. These authors found a series of interesting features in the not yet completely understood dynamics that rules entanglement transfer. By using Wooters concurrence [4] as an entanglement measure, they obtained results that pointed towards the possible existence of conservation laws ruling entanglement transfer, a possible way of achieving entanglement protection. Now if we understand entanglement's evolution of two unconnected parties as a pure information-exchange process, then memory effects become an issue. The system that Yönaç et al analysed was a pair of two-level atoms, each sitting in its own lossless cavity and interacting with a single mode. The dynamics was ruled by the Jaynes–Cummings Hamiltonian. The results presented in [3] suggested that memory effects might play a crucial role and that non-Markovianity could be essential for entanglement's rebirth. As a natural follow-up of these investigations, people addressed similar cases, thereby replacing non-Markovian features by Markovian ones [5–8]. For instance, López et al [5] considered two initially entangled but subsequently unconnected qubits, each of them being in contact with its own environment. The environments were assumed to be reservoirs coupling to the qubit within the Markov approximation. Specifically, the qubits in [5] were single-cavity modes and the reservoirs vacuum states at zero temperature. Under these circumstances, each cavity mode suffers an exponential decay that follows from the Weisskopf–Wigner approximation and the excitation of the cavity is eventually transferred to the environment. Hence, the environment has two possible collective states, one which is empty and a second one hosting a single excited mode within an infinity of other, unoccupied modes. Such a dichotomic situation calls out for a qubit-like description of a system that thereby comprises four qubits, as in [3]. Entanglement can thus be calculated among six possible pairings. López et al concluded that ESD in the two-qubit system was necessarily related to ESB between the respective, memoryless environments [5]. Thus, the conjecture raised in [3], that memory effects might be crucial for entanglement transfer, seemed to be ruled out in [5]. We believe that this conclusion is questionable. Memory effects, like those experimentally tested in [9], appear to be necessary for the recovery of entanglement and further investigation is still needed to settle the issue. Indeed, due to their very nature, reservoirs should be incapable of becoming entangled. To the best of our knowledge, neither a theoretical model nor any experimental observations have been reported about entangled reservoirs. An environment acting as a reservoir must be in a steady state, and it typically has a deleterious effect on quantum information. If results like those reported in [5] were correct, then entangled reservoirs could be used as carriers of quantum information, something that sounds physically meaningless. Now, within the Weisskopf–Wigner approach, the reservoir appears like a qubit only from the system's perspective. Treating the environment as a qubit amounts to its replacement by a so-called environment simulator, a commonly used tool when dealing with open quantum systems [10]. From the system's perspective, there is no difference between the environment and its simulator, as the predicted system's dynamics is the same in both cases. However, physically there is a great difference between a reservoir and a qubit simulating it. To begin with, the jumping from one qubit state to the other is irreconcilable with the defining feature of a reservoir: its being in a steady state. One can certainly implement arrangements behaving like the reservoir's simulators [11], but these arrangements markedly differ from physical reservoirs. For example, Salles et al [12] used dichotomic path modes in an interferometric array that would be suitable for testing entanglement swapping from a system to its environment simulator. However, we cannot take this as a test for predictions about the reservoir's behaviour. In [12], a single photon carries two qubits, one being the system and the other the environment simulator. The latter then constantly jumps between two states, conditioned on what happens to the first. This notoriously differs from the behaviour of a reservoir that—while being itself in steady state—may influence the dynamics of a small system. Clearly, we may not draw conclusions about the environment's behaviour from that of its simulator. The simulator's services finish once we have derived the system's dynamics.

In order to study information transfer, possibly translating into global conservation of quantities like concurrence, we must deal with systems that can be entangled. One such open system, being analytically solvable [10, 13], is a two-level atom in a lossy cavity at zero temperature. It allows us to study the transfer of concurrence among quantal parties, and the irreversible loss of information into a reservoir. By tuning the cavity's quality factor $\mathcal {Q}$, we can go from the underdamped regime (signalled by damped Rabi oscillations) to the overdamped regime (without Rabi oscillations). Qualitatively, the former regime is non-Markovian, while the latter is Markovian. Quantitatively, we can assess the non-Markovianity of quantum processes Φ(t) that connect an initial state—given by the density matrix ρ(0)—with an evolved state ρ(t) = Φ(t)ρ(0) by using a recently introduced measure [14] that depends on the trace distance between two states: D(ρ₁, ρ₂) = tr|ρ₁ − ρ₂|/2, with $\left|A\right|\equiv \sqrt{A^{\dagger }A}$. Using the rate of change σ(t, ρ_{1, 2}(0)) = dD(ρ₁(t), ρ₂(t))/dt for any two initial states ρ_{1, 2}(0), one defines a measure $\mathcal {N}$ for the non-Markovianity of the process Φ as

$$\begin{equation} \mathcal {N}(\Phi )=\max _{\rho _{1,2}(0)}\int _{\sigma >0}\sigma (t,\rho _{1,2}(0))\,{\rm d}t. \end{equation} \tag{ 1 }$$

As explained below, a slightly modified version of this measure proves useful for analysing the connection between non-Markovianity and the dynamics of concurrence. Recent investigations [15, 16] have addressed similar issues and found that in the non-Markovian regime, as expected, a limited amount of information returns to the system. An alternative definition [17] of non-Markovianity is based on the divisibility of the quantum map. The corresponding measure $\mathcal {I}(\Phi )$ is given in terms of the Choi matrix associated with Φ. The condition $\mathcal {I}(\Phi )>0$ signals that the divisibility of a completely positive (CP) map is broken. It has been shown [18] that in some cases $\mathcal {I}(\Phi )>0$ while $\mathcal {N}(\Phi )=0$, the two measures leading thus to conflicting quantifications of the same process, although it holds true [19] that whenever $\mathcal {N}(\Phi )>0,$ $\mathcal {I}(\Phi )> 0$. That is, any process that is diagnosed as non-Markovian according to $\mathcal {N}(\Phi )$, is also non-Markovian according to $\mathcal {I}(\Phi )$, while the converse might not be true. This happens because $\mathcal {N}(\Phi )$ quantifies the backflow of information from the environment to the system, whereas $\mathcal {I}(\Phi )$ is based on a different—though related—feature: the divisibility of the quantum map. Chruściński et al have proposed a way to reconcile the two aforementioned measures [19]. Now, even having the two defining features—divisibility and backflow of information—already captured in a single measure, the question about the connection between non-Markovianity and the dynamics of concurrence would remain still open. We can thus focus on the widely accepted measure $\mathcal {N}(\Phi )$, for the sake of analysing the relationship between non-Markovianity and entanglement protection (nonetheless, in appendix C, we briefly discuss divisibility). We will thus henceforth refer to non-Markovian processes whenever $\mathcal {N}(\Phi )> 0$.

As we shall see, the non-Markovianity of a process does not necessarily preclude a definite loss of information. Indeed, there are processes that are non-Markovian, but in these processes, information could be carried by entanglement and ESD might occur without revival. This raises questions about the meaning of $\mathcal {N}(\Phi )$. Let us recall that the current interpretation of $\mathcal { N}(\Phi )>0$ is as follows. Because there are times during which the distinguishability between two states grows (σ(t, ρ_{1, 2}(0)) > 0), then for these times information is returning to the system. However, we should be careful and not generally ascribe to processes with $\mathcal {N}(\Phi )>0$ the property of being good keepers of information. If we have encoded information using entanglement, then it is its dynamics that matters, not the dynamics of trace distance. We have found σ(t, ρ_{1, 2}(0)) > 0 in cases for which no revival of concurrence takes place. Even more striking, we have found $\mathcal {N}(\Phi )>0$ in purely dissipative, overdamped regimes, which are intuitively taken as Markovian. Thus, an overall diagnosis of Φ as being non-Markovian according to $\mathcal {N}$ not always conveys a useful characterization of the process. This is an important issue, in view of current efforts to counteract information corrupting effects in quantum-information processing.

By addressing the simple, yet not trivial case of a system that consists of two atom–cavity units with losses, we can illustrate the limits of $\mathcal {N}(\Phi )$ as a tool for diagnosing non-Markovianity. The system we address is a realizable one and serves also the purpose of highlighting the difference between a reservoir and its simulator. In the following sections, we present our main results and conclusions. For the sake of clarity, we have relegated the details of some calculations to a series of appendices.

Our basic building block is the density matrix ρ that describes the atom–cavity unit. It evolves in time as ruled by the following equation [10]:

$$\begin{equation} \frac{{\rm d}\rho }{{\rm d}t}=-\frac{\rm i}{\hbar}[H,\rho ]+\frac{\kappa }{2}\left( 2 a\rho a^{\dagger }-a^{\dagger }a\rho -\rho a^{\dagger }a\right)\! , \end{equation} \tag{ 2 }$$

where H = ℏω_egσ_z/2 + ℏω_ca†a − iℏΩ(aσ₊ − a†σ₋)/2. We neglect the atomic spontaneous emission rate Γ_at and assume a cavity's finesse $\mathcal {Q}=\omega _{c}/\kappa \gg \Gamma _{{\rm at}}$. By varying κ and Ω, we can go from the underdamped to the overdamped regime. The atom–photon correlation time is comparable to the atom's evolution time in the underdamped regime, but it becomes negligible in the overdamped case, when the atom is effectively coupled to the reservoir, keeping no memory from past influences. If the initial state has no photons, then the relevant Hilbert space is spanned by |1〉 = |e, 0〉, |2〉 = |g, 1〉, |3〉 = |g, 0〉. Equation (2) can be readily solved, the analytical expressions being simpler for the resonant case, ω_eg = ω_c, that we assume henceforth. As shown in appendix A, the components ρ₁₁, ρ₂₂ and ρ₁₂ + ρ₂₁ are coupled among themselves only, and similarly ρ₁₃, ρ₂₃, which evolve separately from the others (hence also ρ₃₁, ρ₃₂). As for ρ₃₃, it evolves according to dρ₃₃/dt = κρ₂₂ and is thus fixed by ρ₂₂.

We consider now a system comprising two independent atom–cavity units (A, B), and study entanglement between different parties. Taking as the initial state |Ψ_AB(0)〉 = (cos θ|g_A〉|g_B〉 + sin θ|e_A〉|e_B〉)⊗|0_A〉|0_B〉, the corresponding initial density matrix ρ_AB(0) = |Ψ_AB(0)〉〈Ψ_AB(0)| reads

$$\begin{eqnarray} &&\fl \rho _{AB}(0)=\cos ^{2}\theta \left|3,3\right\rangle \left\langle 3,3\right|+\sin ^{2}\theta \left|1,1\right\rangle \left\langle 1,1\right|\nonumber\\ &&+\, \cos \theta \sin \theta \left[ \left|3,3\right\rangle \left\langle 1,1\right|+\left|1,1\right\rangle \left\langle 3,3\right|\right] , \end{eqnarray} \tag{ 3 }$$

with |3, 3〉〈1, 1| = |3_A〉〈1_A|⊗|3_B〉〈1_B|, etc. The general solution of equation (2) yields the evolution of each atom–cavity subsystem: ρ_{i = A, B}(t), thereby defining the map ρ_i(0) → ρ_i(t) = ∑_μK_(i)μ(t)ρ_i(0)K†_(i)μ(t), with the Kraus operators K_(i)μ. The evolution of ρ_AB is given by ρ_AB(0) → ρ_AB(t) = ∑_μνK_(A)μ(t)⊗K_(B)ν(t)ρ_AB(0)K†_(B)ν(t)⊗K†_(A)μ(t). Thus, by applying this last map to each state in equation (3), we obtain ρ_AB(t) from ρ_AB(0). Using the general solution of equation (2), one obtains |3〉〈3| → |3〉〈3|; |3〉〈1| → ρ₃₁(t)|3〉〈1| + ρ₃₂(t)|3〉〈2|; |1〉〈1| → ρ₁₁(t)|1〉〈1| + ρ₁₂(t)|1〉〈2| + ρ₂₁(t)|2〉〈1| + ρ₂₂(t)|2〉〈2| + ρ₃₃(t)|3〉〈3|, from which ρ_AB(t) follows, as shown in appendix B. Once we have ρ_AB(t), we can study different partitions of our system. A key parameter is α² ≡ κ² − 4Ω². Depending on whether α²≶0, we are in the underdamped or in the purely dissipative regime, as further explained in appendix A.1. As we shall see, for positive values of α², entanglement asymptotically decays. This would be consistent with a concomitant Markovian evolution. To see whether this is the case, we evaluate $\mathcal {N}$ for the quantum maps that correspond to different partitions of our system. Now, the evaluation of $\mathcal {N}$, as defined in equation (1), requires that we perform a maximization over all initial states. For our present purposes, however, we may restrict the maximization to a subset of initial states. We will comment on this point later on but let us now stress that, in practice, when performing a quantum computational task, one is generally restricted to deal with a subset of initial states.

We thus define a restricted non-Markovianity measure $\mathcal {N}_{r}\left( \Phi _{aa}\right)$ similarly to $\mathcal {N}\left( \Phi \right)$ in equation (1), but with the maximization restricted to the initial values ρ_aa(θ, 0) of a given form, e.g., equation (3) traced over the cavities. This is a valid definition, with the proviso that it does not characterize the whole map Φ_aa, only its restriction to a well-defined domain $\mathcal {D}_{\theta }\ni \rho _{aa}(\theta ,0)$. Note that if there are two initial states in $\mathcal {D}_{\theta }$ (labelled θ₁, θ₂) and a time t for which σ_aa(t, θ₁, θ₂) > 0, then Φ_aa is non-Markovian according to the standard definition. On the other hand, according to this definition, the vanishing of $\mathcal { N}_{r}\left( \Phi _{aa}\right)$ for all initial states in $\mathcal { D}_{\theta }$ is not sufficient for Φ_aa to be Markovian (it should be $\mathcal {N} \left( \Phi _{aa}\right) =0$). It is nonetheless useful to term Φ_aa a (restricted) Markovian map, whenever $\mathcal {N}_{r}\left( \Phi _{aa}\right) =0$.

In the following, we will assess the non-Markovianity of different partitions and compare them with the corresponding concurrences. Generally, calculating $\mathcal {N}$ requires evaluation of ρ_d ≡ ρ₁ − ρ₂, which is Hermitian, so that $\left|\rho _{d}\right|= \sqrt{\rho _{d}^{2}}$ and $D\left( \rho _{1},\rho _{2}\right) =\frac{1}{2} {\rm tr}\left|\rho _{d}\right|=\frac{1}{2}\sum _{k}\left|r_{k}\right|$, where the r_k are the eigenvalues of ρ_d. We thus obtain for the rate of change σ(t, ρ_{1, 2}(0)) = dD(ρ₁, ρ₂)/dt the expression $\sigma \left( t,\rho _{1,2}(0)\right) =\frac{1}{2}\sum _{k}{\rm sgn}(r_{k})\,{\rm d}r_{k}/{\rm d}t$, which yields analytical results.

2.1. The atom–atom partition

Let us first address the atom–atom partition. Tracing ρ_AB(t) over the cavities gives the atom–atom density matrix

$$\begin{equation} \rho _{aa}(\theta ,t)=\left( \begin{array}{@{}cccc@{}}c^{2} + s ^{2} \chi ^{2} & 0 & 0 & c s \rho _{31}^{2} \\ 0 & s ^{2}\zeta & 0 & 0 \\ 0 & 0 & s ^{2} \zeta & 0 \\ c s \rho _{13}^{2} & 0 & 0 & s^{2} \rho _{11}^{2} \end{array} \right)\!, \end{equation} \tag{ 4 }$$

with c = cos θ, s = sin θ, χ = ρ₂₂ + ρ₃₃, ζ = ρ₁₁χ. The process Φ_aa: ρ_aa(0) → ρ_aa(t) that is defined through equation (4) gives ρ^dif_aa(t) = ρ_aa(θ₂, t) − ρ_aa(θ₁, t) as

$$\begin{equation} \rho _{aa}^{\rm dif}(t)=\frac{1}{2}\left( \begin{array}{@{}cccc@{}}-c_{d}\xi & 0 & 0 & s_{d}\rho _{31}^{2} \\ 0 & -c_{d}\mu & 0 & 0 \\ 0 & 0 & -c_{d}\mu & 0 \\ s_{d}\rho _{13}^{2} & 0 & 0 & -c_{d}\rho _{11}^{2} \end{array} \right) \!, \end{equation} \tag{ 5 }$$

with c_d = cos 2θ₂ − cos 2θ₁, s_d = sin 2θ₂ − sin 2θ₁, ξ = ρ₁₁(ρ₁₁ − 2), μ = ρ₁₁(1 − ρ₁₁). In order to obtain $\mathcal {N}_{r}\left( \Phi _{aa}\right),$ we need the eigenvalues of ρ^dif_aa(t). These are

$$\begin{equation} r_{1} =r_{2}=-\frac{c_{d}}{2}\left( 1-\rho _{11}\right) \rho _{11}, \end{equation} \tag{ 6a }$$

$$\begin{equation} r_{3} =\frac{1}{2}\big( c_{d}( 1-\rho _{11}) \rho _{11}-\sqrt{ c_{d}^{2}\rho _{11}^{2}+s_{d}^{2}|\rho _{13}|^{4}} \big) , \end{equation} \tag{ 6b }$$

$$\begin{equation} r_{4} =\frac{1}{2}\big( c_{d}( 1-\rho _{11}) \rho _{11}+\sqrt{ c_{d}^{2}\rho _{11}^{2}+s_{d}^{2}|\rho _{13}|^{4}} \big) . \end{equation} \tag{ 6c }$$

Since c²_d(2 − ρ₁₁)ρ³₁₁ + s_d²|ρ₁₃|⁴ ⩾ 0, it follows that c²_d(1 − ρ₁₁)²ρ²₁₁ = c_d²ρ²₁₁ − c_d²(2 − ρ₁₁)ρ³₁₁ ⩽ c²_dρ₁₁² + s²_d|ρ₁₃|⁴, so that

$$\begin{equation} \left|c_{d}\right|(1-\rho _{11})\rho _{11}\le \sqrt{c_{d}^{2}\rho _{11}^{2}+s_{d}^{2}\left|\rho _{13}\right|^{4}} . \end{equation} \tag{ 7 }$$

We can see that r₃ ⩽ 0. Indeed, for c_d < 0 it is obvious, and for c_d > 0 the inequality (7) implies r₃ ⩽ 0. Similarly, one obtains r₄ ⩾ 0. From the explicit results of appendix A, we can check that |ρ₁₃|² = ρ₁₁, so that we have

$$\begin{eqnarray*} \left|r_{1}\right|&=&\left|r_{2}\right|=\frac{ \left|c_{d}\right|}{2}(1-\rho _{11})\rho _{11}, \\ \left|r_{3}\right|&=&-r_{3}=-\frac{c_{d}}{2}(1-\rho _{11})\rho _{11}+\frac{1}{2}\sqrt{c_{d}^{2}+s_{d}^{2}}\rho _{11}, \\ \left|r_{4}\right|&=&r_{4}=\frac{c_{d}}{2}(1-\rho _{11})\rho _{11}+\frac{1}{2}\sqrt{c_{d}^{2}+s_{d}^{2}}\rho _{11}. \end{eqnarray*}$$

We can now calculate the trace distance as $D_{aa}( \rho _{1},\rho _{2}) =\frac{1}{2}\sum _{k=1}^{4}|r_{k}|=\frac{1}{2 }\big(|c_{d}|(1-\rho _{11})+\sqrt{c_{d}^{2}+s_{d}^{2}} \big) \rho _{11}$ and its rate of change is thus given by

$$\begin{equation*} \sigma _{aa}=\frac{{\rm d}D_{aa}}{{\rm d}t}=\frac{1}{2}\big(|c_{d}|(1-2\rho _{11})+ \sqrt{c_{d}^{2}+s_{d}^{2}}\big) \frac{{\rm d}\rho _{11}}{{\rm d}t}. \end{equation*}$$

Because $2\rho _{11}\le 2\le 1+\sqrt{1+\left( s_{d}/c_{d}\right) ^{2}}$, it follows that $\left|c_{d}\right|(1-2\rho _{11})+\sqrt{ c_{d}^{2}+s_{d}^{2}}\ge 0$, so that dD_aa/dt > 0⇔dρ₁₁/dt > 0 . The explicit expression of dρ₁₁(t)/dt reads

$$\begin{eqnarray*} &&\fl \frac{{\rm d}\rho _{11}}{{\rm d}t}=\frac{\Omega ^2 {\rm e}^{-\frac{t \kappa }{2}}}{\alpha ^2}\left[\kappa \left(1-\cosh \left(\frac{1}{2} t \sqrt{\alpha ^2}\right)\right)\right.\nonumber\\ &&\left.-\, \sqrt{\alpha ^2} \sinh \left(\frac{1}{2} t \sqrt{\alpha ^2}\right)\right]\!. \end{eqnarray*}$$

One sees that for α² > 0, it holds dρ₁₁/dt < 0, so that the evolution of ρ_aa(θ, t) is indeed Markovian in this regime.

For α² < 0, there are time intervals for which σ_aa > 0, as illustrated in figure 1, and the process is thus non-Markovian. From the given expression of dρ₁₁/dt, we can obtain the times t_m for which σ_aa = 0. The time intervals for which σ_aa > 0 are [t₁, t₂], [t₃, t₄], ..., so that we can write

$$\begin{eqnarray*} &&\fl \int _{\sigma _{aa}>0}\sigma _{aa} \,{\rm d}t=(D_{aa}(t_{2})-D_{aa}(t_{1}))+(D_{aa}(t_{4})-D_{aa}(t_{3}))\nonumber\\ &&+\, \dots =\sum _{k=1}^{\infty }D_{aa}(t_{2k})-\sum _{k=0}^{\infty }D_{aa}(t_{2k+1}). \end{eqnarray*}$$

After some algebra, we obtain

$$\begin{equation*} t_{m}=\frac{2m\pi }{\sqrt{4\Omega ^{2}-\kappa ^{2}}}, \ \ \ \ m=1,2,\ldots . \end{equation*}$$

Now, because

$$\begin{equation*} 2D_{aa}(t)=\big( |c_{d}|+\sqrt{c_{d}^{2}+s_{d}^{2}}\big) \rho _{11}(t)-|c_{d}|\rho _{11}^{2}(t), \end{equation*}$$

in order to calculate $\int _{\sigma _{aa}>0}\sigma _{aa} \,{\rm d}t$ we need to evaluate only the quantities I₁ = ∑^∞_{k = 1}ρ₁₁(t_2k) − ∑^∞_{k = 0}ρ₁₁(t_{2k + 1}) and I₂ = ∑^∞_{k = 1}ρ²₁₁(t_2k) − ∑^∞_{k = 0}ρ₁₁²(t_{2k + 1}). We obtain the following closed expressions:

$$\begin{eqnarray*} &&\fl I_{1}=\sum _{j=1}^{\infty }{\rm e}^{-2{\rm j}\pi /\sqrt{\Omega _{r}^{2}-1}}\left[ 1-\frac{{\rm e}^{\pi /\sqrt{\Omega _{r}^{2}-1}}}{\Omega _{r}^{2}-1}\right] \nonumber\\ &&\hphantom{\fl I_{1}}=\frac{1}{2}\left[ 1-\frac{{\rm e}^{\pi /\sqrt{\Omega _{r}^{2}-1 }}}{\Omega _{r}^{2}-1}\right] \left[ \coth \left( \frac{\pi }{\sqrt{\Omega _{r}^{2}-1}}\right) -1\right] \!, \nonumber\\ &&\fl I_{2}=\sum _{j=1}^{\infty }{\rm e}^{-4{\rm j}\pi /\sqrt{ \Omega _{r}^{2}-1}}\left[ 1-\frac{{\rm e}^{2\pi /\sqrt{\Omega _{r}^{2}-1}}}{\left( \Omega _{r}^{2}-1\right) ^{2}}\right] \nonumber\\ &&\hphantom{\fl I_{1}}=\frac{1}{2}\left[ 1-\frac{{\rm e}^{2\pi / \sqrt{\Omega _{r}^{2}-1}}}{\left( \Omega _{r}^{2}-1\right) ^{2}}\right] \left[ \coth \left( \frac{2\pi }{\sqrt{\Omega _{r}^{2}-1}}\right) -1\right]\! , \end{eqnarray*}$$

where Ω_r = 2Ω/κ. Note that the positiveness of the above expressions requires Ω_r to be constrained by Ω_r ⩾ 2.3325, which follows from $\exp (\pi /\sqrt{\Omega _{r}^{2}-1})\le \Omega _{r}^{2}-1$. In any case, it is clear that $\mathcal {N} _{r}=\max _{\rho _{1,2}(0)}\int \sigma _{aa} \left( t,\rho _{1,2}(0)\right) \,{\rm d}t$ is a quantity that scales as $\mathcal {N}_{r}\left( \Omega /\kappa \right)$.

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The behaviour of σ_aa(t), as shown in figure 1 for the underdamped case, is similar to that of concurrence in the same regime. We could thus also talk of sudden death and rebirth in terms of σ_aa(t). We will return to this point and compare concurrence and σ_aa(t) in a regime in which both behave similarly.

2.2. The cavity–cavity partition

By applying a similar procedure, we obtain for the cavity–cavity subsystem a matrix ρ_cc which follows from ρ_aa by the replacements $\rho _{11}\leftrightarrows \rho _{22}$, ρ₃₁ → ρ₃₂, ρ₁₃ → ρ₂₃. From the so-obtained ρ_cc, it follows that ρ^dif_cc(t) = ρ_cc(θ₂, t) − ρ_cc(θ₁, t) is given by

$$\begin{equation} \rho _{cc}^{\rm dif}(t)=\frac{1}{2}\left( \begin{array}{@{}cccc@{}}-c_d\nu & 0 & 0 & s_d\rho _{32}^{2} \\ 0 & -c_d\varpi & 0 & 0 \\ 0 & 0 & -c_d\varpi & 0 \\ s_d\rho _{23}^{2} & 0 & 0 & -c_d\rho _{22}^{2} \end{array} \right) \!, \end{equation} \tag{ 8 }$$

with c_d = cos 2θ₂ − cos 2θ₁, s_d = sin 2θ₂ − sin 2θ₁, ν = ρ₂₂(ρ₂₂ − 2) and ϖ = ρ₂₂(1 − ρ₂₂). As in the case of the atom–atom subsystem, for calculating $\mathcal {N}_{r}\left( \Phi _{cc}\right)$ we need the eigenvalues of ρ^dif_cc(t), which are

$$\begin{equation} r_{1} =r_{2}=-\frac{c_{d}}{2}\left( 1-\rho _{22}\right) \rho _{22}, \end{equation} \tag{ 9a }$$

$$\begin{equation} r_{3} =\frac{1}{2}\big( c_{d}( 1-\rho _{22}) \rho _{22}-\sqrt{ c_{d}^{2}\rho _{22}^{2}+s_{d}^{2}|\rho _{23}|^{4}} \big) , \end{equation} \tag{ 9b }$$

$$\begin{equation} r_{4} =\frac{1}{2}\big( c_{d}( 1-\rho _{22}) \rho _{22}+\sqrt{ c_{d}^{2}\rho _{22}^{2}+s_{d}^{2}|\rho _{23}|^{4}} \big) . \end{equation} \tag{ 9c }$$

From the explicit solution given in appendix A, we observe that |ρ₂₃|² = ρ₂₂, and by a similar reasoning as in the previous section, one obtains r₃ ⩽ 0 and r₄ ⩾ 0, so that the trace distance results in

$$\begin{eqnarray*} &&\fl D_{cc}( \rho _{1},\rho _{2}) =\frac{1}{2}\sum _{k=1}^{4}|r_{k}|=\frac{1}{2} \big( |c_{d}|(1-\rho _{22})+\sqrt{c_{d}^{2}+s_{d}^{2}} \big) \rho _{22}. \end{eqnarray*}$$

Its rate of change is then given by

$$\begin{equation*} \sigma _{cc}=\frac{{\rm d}D_{cc}}{{\rm d}t}=\frac{1}{2}\big( |c_{d}|(1-2\rho _{22})+ \sqrt{c_{d}^{2}+s_{d}^{2}}\big) \frac{{\rm d}\rho _{22}}{{\rm d}t}. \end{equation*}$$

As with the atom–atom evolution, also in this case we easily conclude that $\left|c_{d}\right|(1-2\rho _{22})+\sqrt{ c_{d}^{2}+s_{d}^{2}}\ge 0$, so that in order to see whether dD_cc/dt > 0 we need to analyse only the sign of

$$\begin{equation*} \frac{{\rm d}\rho _{22}}{{\rm d}t}=\frac{2\Omega ^2 {\rm e}^{-t\kappa /2}}{\left(\alpha ^2\right)^{3/2}} \sinh \left(\frac{t}{4}\sqrt{\alpha ^2}\right)\eta , \end{equation*}$$

where $\eta = \alpha ^2 \cosh (t \sqrt{\alpha ^2}/4) - \kappa \sqrt{\alpha ^2} \sinh (t \sqrt{\alpha ^2}/4)$. This expression can be written in the form

$$\begin{equation*} \frac{\eta }{2\Omega \sqrt{\alpha ^2}} = \frac{\sqrt{\alpha ^2}}{2\Omega } \cosh \left(\frac{t}{4}\sqrt{\alpha ^2}\right) - \frac{\kappa }{2\Omega } \sinh \left(\frac{t}{4}\sqrt{\alpha ^2}\right)\!. \end{equation*}$$

By setting cosh u = κ/2Ω and $\sinh u = \sqrt{\alpha ^2}/2\Omega ,$ the above equation reads

$$\begin{equation*} \frac{\eta }{2\Omega \sqrt{\alpha ^2}} = -\sinh \left(\frac{t}{4}\sqrt{\alpha ^2}-u\right)\!, \end{equation*}$$

with

$$\begin{equation*} u = \tanh ^{-1}\left(\frac{\sqrt{\alpha ^2}}{\kappa }\right)\!. \end{equation*}$$

For α² > 0, we see that η > 0 and thus σ_cc > 0 provided that t is such that $t\sqrt{\alpha ^2}/4<u$. In this case, the evolution is non-Markovian, even though we are in a purely dissipative regime. We see also that there is a single time t₀ for which σ_cc = 0, which is given by

$$\begin{equation} t_{0}=\frac{4}{\sqrt{\alpha ^{2}}}\tanh ^{-1}\left(\frac{\sqrt{\alpha ^{2}}}{ \kappa }\right)\!. \end{equation} \tag{ 10 }$$

The behaviour of t₀(κ, Ω) is shown in figure 2. We will come back to discuss the behaviour of t₀(κ, Ω), after we have dealt with concurrences.

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For α² < 0, the evolution is also non-Markovian, as expected, but now the times t_m for which σ_cc = 0 are given by

$$\begin{equation*} t_{m}=\frac{4}{\sqrt{4\Omega ^{2}-\kappa ^{2}}}\left(m\pi +\cos ^{-1}\left(\frac{\kappa }{2\Omega }\right)\right), \ \ \ \ m=0,1,2,\ldots . \end{equation*}$$

By proceeding similarly to the atom–atom case, one can calculate $\mathcal {N}_{r}$, if needed.

In the present case, for all the two-qubit partitions of ρ_AB, we obtain matrices of the X form [3]. Concurrences are then given by $C_{jk}(\rho )=2 \max \lbrace 0,\left|\rho _{14}\right|-\sqrt{\rho _{22} \rho _{33}},\left|\rho _{23}\right|-\sqrt{\rho _{11} \rho _{44}}\rbrace$, with j, k ∈ {a, c} (we append a subindex if necessary, to avoid ambiguity: a_Ac_B means atom of unit A and cavity of unit B, etc). In the present case, we obtain C_jk = 2max {0, Q_jk}, with

$$\begin{eqnarray*} &&\fl Q_{aa} = \left|\cos \theta \sin \theta \right|\rho _{11}-\sin ^{2}\theta \left( 1-\rho _{11}\right)\rho _{11}, \nonumber \end{eqnarray*}$$

$$\begin{eqnarray*} &&\fl Q_{cc} = \left|\cos \theta \sin \theta \right|\rho _{22}-\sin ^{2}\theta \left( 1-\rho _{22}\right)\rho _{22}, \nonumber \end{eqnarray*}$$

$$\begin{eqnarray*} &&\fl Q_{a_{A}c_{B}} = Q_{a_{B}c_{A}} \nonumber\\ &&= (|\cos \theta \sin \theta |-\sqrt{(1-\rho _{11})(1-\rho _{22})}\sin ^{2}\theta)\sqrt{\rho _{11}\rho _{22}}, \end{eqnarray*}$$

$$\begin{eqnarray} &&\fl Q_{a_{A}c_{A}} = Q_{a_{B}c_{B}}= \sin ^{2}\theta \sqrt{\rho _{11}\rho _{22}}. \end{eqnarray} \tag{ 11 }$$

Concurrences of all partitions are thus ruled by just two quantities, ρ₁₁ and ρ₂₂, the populations of |e, 0〉 and |g, 1〉. Based on equations (11), we can address questions like those raised by Yönaç et al [3]. Consider, e.g., C_aa and C_cc. We see from equations (11) that the conditions C_aa > 0 and C_cc > 0 are equivalent to f(ρ₁₁, θ) > 0 and f(ρ₂₂, θ) > 0, respectively, with

$$\begin{equation} f(x,\theta )= x+|\cot \theta |-1. \end{equation} \tag{ 12 }$$

Thus, by analysing f(x, θ) we can predict ESD, ESB and entanglement transfer. An example is shown in figure 3, in which we observe entanglement transfer as well as entanglement rebirth in the underdamped regime (α² < 0), signalled by the atom–atom and cavity–cavity concurrences. On the other hand, as illustrated in figure 4, in the purely dissipative regime (α² > 0), neither ESD nor ESB shows up. Both atom–atom and cavity–cavity concurrences asymptotically decay. However, a non-vanishing C_cc, signalling entanglement transfer, is somewhat unexpected for cavities that behave as reservoirs for the atoms. Clearly, the collective behaviour does not reflect the behaviour of the constituent parties. In any case, we should expect that under these circumstances, the evolution of the corresponding partition of the system is Markovian. To confirm this, we look at $\mathcal {N}_{r}$ for the respective maps.

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3.1. Interplay between entanglement and non-Markovianity

As already shown, for the atom–atom partition we have σ_aa(t, θ₁, θ₂) > 0⇔dρ₁₁/dt > 0, while for the cavity–cavity partition σ_cc(t, θ₁, θ₂) > 0⇔dρ₂₂/dt > 0. Within the purely dissipative regime (α² > 0), it holds dρ₁₁/dt < 0, so that in the atom–atom case $\mathcal { N}_{r}\left( \Phi _{aa}\right)=0$, as expected. However, there are cases (see figure 5, left panel) in which σ_cc(t, θ₁, θ₂) > 0 for 0 < t < t₀. This implies that for the cavity–cavity partition, the process Φ_cc can be diagnosed as non-Markovian ($\mathcal { N}_{r}\left( \Phi _{cc}\right)>0$), even within a purely dissipative regime (α² > 0).

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Furthermore, ρ^dif_aa yields results like that of figure 5, right panel. We observe sudden death and rebirth in both C_aa(t, θ₁) and σ_aa(t, θ₁, θ₂) > 0, the latter being required for having $\mathcal { N}_{r}\left( \Phi _{aa}\right)>0$. As we see, sudden death and rebirth in one and the other case obey different dynamics. Even when C_aa(t, θ) is definitely gone, σ_aa(t, θ, −θ) keeps reappearing during time windows, each window lasting a time T_aa = 2π(4Ω² − κ²)^−1/2. The rate of change σ_aa(t, θ, −θ) corresponds to the distance between two entangled states having the same concurrence. If we employ atom–atom entangled states as a resource, and if at a given time this resource becomes unavailable, it is immaterial whether two such evolved states remain distinguishable. Hence, figure 5 illustrates how a diagnosis about the return of 'information' back to the system (σ_aa > 0) fails to capture that our chosen carrier of information, namely entanglement, is not available any longer (C_aa = 0). The non-Markovianity of this map can be calculated, giving $\mathcal {N}_{r}\left( \Phi _{aa}\right)>0$ and is therefore somewhat misleading for assessing the action of Φ_aa. Similar remarks can be made for the map Φ_cc.

3.2. Additional entanglement features in the system

Besides α, also Γ = Ω²/κ is a physically meaningful parameter. For large enough α, the excitation of each atom–cavity unit exponentially decays ruled by Γ alone, as shown in appendix A.1. Thus, Γ serves us to make sure that we are definitely in a regime in which each atom sees its cavity as a reservoir. In such a case, we should expect that the cavities are unable to entangle. However, for appropriate choices of Ω and κ, we find non-vanishing cavity–cavity concurrences which initially grow, reach a maximum and finally decay approaching zero asymptotically. This was, for instance, the case shown in figure 4. It illustrates a rather counterintuitive feature, because it signals entanglement transfer from qubits to entities which from the qubit's perspective look like reservoirs.

On the other hand, as we saw before, for α² > 0 there is a single time t₀, given by equation (10), for which σ_cc = 0. We can study how t₀(κ, Ω) changes when we vary Ω and κ, keeping Γ = Ω²/κ fixed. We can further choose κ ≫ Ω, so as to be in the overdamped regime, in which the atomic dynamics is ruled by Γ alone. The cavities are then practically all the time in their vacuum states and behave as reservoirs for the atoms. We should thus expect that within this regime, no quantum features related to the cavities show up. But this is not the case. As shown in figure 6, t₀ is not a function of Γ alone. It depends separately on κ and Ω, so that for a given Γ we can obtain quite different values of t₀. We can thus identify here a feature that is related to the entanglement of two cavities, even though we should expect these cavities to be incapable of becoming entangled, as they behave like reservoirs for the individual atoms. This is reminiscent of ESD: the collective dynamics of the entangled system markedly departs from the dynamics of its constituents. Furthermore, such a collective dynamics is established in the absence of any physical connection between the constituents. In contrast to other studies that address (quantum) simulators, the effects reported here involve quantal environments that can effectively behave like reservoirs for some other entities, while keeping their entanglement's capability.

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3.3. Experimental feasibility

In this paper, we have dealt with one of the few open systems that can be exactly solved and that is capable of showing Markovian and non-Markovian regimes: a two-level atom in a lossy cavity. By taking two such atom–cavity units to conform a larger system, it was possible to analytically study its evolution when only the atoms are initially entangled. We could observe entanglement transfer under Markovian and non-Markovian regimes. The lossy cavities may behave either as quantum objects or as reservoirs, depending on the chosen regime. It is possible to observe entanglement transfer from the atoms to the cavities. The latter represent actual physical systems. Our findings refer to these actual systems, not to their simulators. In particular, we have predicted entanglement transfer when the cavities appear to the atoms as reservoirs. This phenomenon could in principle be submitted to experimental test and contrasted against recently studied cases in which actual reservoirs were substituted by (quantum) simulators [12]. Our results confirm also a recent claim [25], namely that the generation of bright and dark periods in the entanglement dynamics can occur for a wide variety of environments.

In order to quantitatively assess the degree of non-Markovianity, we applied a recently introduced measure $\mathcal {N}(\Phi )$ that is based on the trace distance between states. We have submitted this measure to critical scrutiny. The current interpretation of a growing distance between states is that 'information' is flowing from the environment back to the system. We argue that it is necessary to tighten this interpretation by being more specific about the chosen carriers of information. It might be that a quantum process is classified as non-Markovian, but information could have been encoded using a feature not directly related to the distinguishability between states. In order to illustrate the possible shortcomings of $\mathcal {N}(\Phi ),$ we compared it with the degree of entanglement, as measured by Wooters concurrence. We have presented a quantum map for which σ_aa(t) > 0 when concurrence—a possible carrier of information—is no longer available (C_aa(t) = 0). Moreover, we have found (figure 5, left panel) that $\mathcal {N}(\Phi )>0$ in purely dissipative, overdamped regimes, which are intuitively taken as Markovian.

Finally, we stress that by having restricted our analysis to a specific family of entangled states, we followed an approach that is representative of most quantum informational tasks. Any source of entanglement defines a set of available initial states, thereby fixing an effective and restricted quantum channel. Our analysis could be extended to different families of initial states. Future work might show interesting features connected to the chosen family, but the general picture should prevail: generally, we should expect that entanglement can be transferred from a system only when its environment is capable of carrying quantum information. In our model, entanglement occurs between quantal environments, not between reservoirs. Our treatment thus avoids paradoxical situations involving entangled reservoirs, something reminiscent of Schrödinger's cat. We emphasize the inadequacy of using environment simulators as substitutes of true environments when the dynamics of the latter matters. Also, that proper non-Markovianity measures should be tailored to the actually employed carriers of information. To find such measures is thus a challenge issued by our results.

We thank B Mejía for critical reading of our manuscript, and DGI-PUCP for partial financial support.

The Lindblad equation for the atom–cavity unit yields the following differential equations for the components of ρ:

$$\begin{eqnarray} &&\fl \frac{\rm d}{{\rm d}t}\left( \begin{array}{@{}c@{}}\rho _{11} \\ \rho _{22} \\ \rho _{12}+\rho _{21} \end{array} \right) =\left( \begin{array}{@{}ccc@{}}0 & 0 & -\Omega /2 \\ 0 & -\kappa & \Omega /2 \\ \Omega & -\Omega & -\kappa /2 \end{array} \right) \left( \begin{array}{@{}c@{}}\rho _{11} \\ \rho _{22} \\ \rho _{12}+\rho _{21} \end{array} \right) , \nonumber\\ \end{eqnarray} \tag{ A.1 }$$

$$\begin{eqnarray} &&\fl \frac{\rm d}{{\rm d}t}(\rho _{12}-\rho _{21})=-\frac{\kappa }{2}(\rho _{12}-\rho _{21}), \end{eqnarray} \tag{ A.2 }$$

$$\begin{eqnarray} &&\fl \frac{{\rm d}\rho _{33}}{{\rm d}t}=\kappa \rho _{22}, \end{eqnarray} \tag{ A.3 }$$

$$\begin{eqnarray} &&\fl \frac{\rm d}{{\rm d}t}\left( \begin{array}{@{}c@{}}\rho _{13} \\ \rho _{23} \end{array} \right)=\left( \begin{array}{@{}cc@{}}-{\rm i}\omega & -\Omega /2 \\ \Omega /2 & -{\rm i}\omega -\kappa /2 \end{array} \right)\left( \begin{array}{@{}c@{}}\rho _{13} \\ \rho _{23} \end{array} \right)\!, \end{eqnarray} \tag{ A.4 }$$

$$\begin{eqnarray} &&\fl \frac{\rm d}{{\rm d}t}\left( \begin{array}{@{}c@{}}\rho _{31} \\ \rho _{32} \end{array} \right) =\left( \begin{array}{@{}cc@{}}{\rm i}\omega & -\Omega /2 \\ \Omega /2 & {\rm i}\omega -\kappa /2 \end{array} \right) \left( \begin{array}{@{}c@{}}\rho _{31} \\ \rho _{32} \end{array} \right) \!. \end{eqnarray} \tag{ A.5 }$$

The system of equations (A.1)–(A.5) can be explicitly solved, thereby defining a map

$$\begin{equation*} \rho (t)=\Phi \left( \rho (0)\right) =\sum _{i}K_{i}(t)\rho (0)K_{i}^{\dagger }(t). \end{equation*}$$

For an initial state ρ(0) = ∑_{k, l}α_kl|k〉〈l|, we obtain ρ(t) from the action of K_i(t) on each element |k〉〈l|, i.e.

$$\begin{equation*} \left|k\right\rangle \left\langle l\right|\rightarrow K_{i}(t)\left|k\right\rangle \left\langle l\right|K_{i}^{\dagger }(t). \end{equation*}$$

The action of the map on |k〉〈l| can be obtained case by case. First, from equations (A.1)–(A.5), the evolution of |3〉〈3| follows from setting ρ₃₃(0) = 1 and all other ρ_ij(0) = 0. This gives (see equations (A.1) and (A.3)) ρ₃₃(t) = ρ₃₃(0) = 1, so that

$$\begin{equation*} \left|3\right\rangle \left\langle 3\right|\rightarrow \left|3\right\rangle \left\langle 3\right|. \nonumber \end{equation*}$$

The evolution of |1〉〈3| and |3〉〈1| follows from setting ρ₁₃(0) = 1 = ρ₃₁(0) and all other ρ_ij(0) = 0. We obtain

$$\begin{eqnarray*} \left|3\right\rangle \left\langle 1\right|& \rightarrow \rho _{31}(t)\left|3\right\rangle \left\langle 1\right|+\rho _{32}(t)\left|3\right\rangle \left\langle 2\right|, \nonumber \end{eqnarray*}$$

$$\begin{eqnarray*} \left|1\right\rangle \left\langle 3\right|& \rightarrow \rho _{13}(t)\left|1\right\rangle \left\langle 3\right|+\rho _{23}(t)\left|2\right\rangle \left\langle 3\right|, \nonumber \end{eqnarray*}$$

where

$$\begin{eqnarray*} &&\fl \rho _{31}(t) =\exp \left( -\frac{t}{4}\left( \kappa -4{\rm i}\omega \right) \right) \nonumber\\ &&\times \left( \cosh \left( \frac{t}{4}\sqrt{\alpha ^{2}}\right) +\frac{ \kappa \sinh \left( \frac{t}{4}\sqrt{\alpha ^{2}}\right) }{\sqrt{\alpha ^{2}} }\right)\! , \\ &&\fl \rho _{32}(t) =\frac{2\Omega \exp \left( -\frac{t}{4}\left( \kappa -4{\rm i}\omega \right) \right) }{\sqrt{\alpha ^{2}}}\sinh \left( \frac{t}{4}\sqrt{ \alpha ^{2}}\right) \!. \end{eqnarray*}$$

For the evolution of |1〉〈1|, we set ρ₁₁(0) = 1, all other ρ_ij = 0, and obtain

$$\begin{eqnarray*} &&\fl \left|1\right\rangle \left\langle 1\right|\rightarrow \rho _{11}(t)\left|1\right\rangle \left\langle 1\right|+\rho _{12}(t)\left|1\right\rangle \left\langle 2\right|\nonumber\\ &&+\,\rho _{21}(t)\left|2\right\rangle \left\langle 1\right|+\rho _{22}(t)\left|2\right\rangle \left\langle 2\right|+\rho _{33}(t)\left|3\right\rangle \left\langle 3\right|, \end{eqnarray*}$$

with

$$\begin{eqnarray*} &&\fl \rho _{11}(t) =\frac{{\rm e}^{-t\kappa /2}}{\alpha ^{2}}\Bigg( \kappa \sqrt{ \alpha ^{2}}\sinh \Bigg( \frac{t}{2}\sqrt{\alpha ^{2}}\Bigg) \nonumber\\ &&+\, ( \kappa ^{2}-2\Omega ^{2}) \cosh \Bigg( \frac{t}{2}\sqrt{\alpha ^{2}}\Bigg) -2\Omega ^{2}\Bigg) , \\ &&\fl \rho _{12}(t) =\frac{2\Omega {\rm e}^{-t\kappa /2}}{\alpha ^{2}}\sinh \Bigg( \frac{t}{4}\sqrt{\alpha ^{2}}\Bigg) \Bigg( \sqrt{\alpha ^{2}}\cosh \Bigg( \frac{t}{4}\sqrt{\alpha ^{2}}\Bigg) \nonumber\\ &&+\, \kappa \sinh \Bigg( \frac{t}{4}\sqrt{ \alpha ^{2}}\Bigg) \Bigg) , \\ &&\fl \rho _{21}(t) =\rho _{12}(t), \\ &&\fl \rho _{22}(t) =\frac{4\Omega ^{2}{\rm e}^{-t\kappa /2}}{\alpha ^{2}}\sinh ^{2}\Bigg( \frac{t\sqrt{\alpha ^{2}}}{4}\Bigg) , \\ &&\fl \rho _{33}(t) =1+\frac{{\rm e}^{-t\kappa /2}}{\alpha ^{2}}\Bigg[ 4\Omega ^{2}-\kappa \Bigg( \sqrt{\alpha ^{2}}\sinh \Bigg( \frac{t}{2}\sqrt{\alpha ^{2}}\Bigg) \nonumber\\ &&+\, \kappa \cosh \Bigg( \frac{t}{2}\sqrt{\alpha ^{2}}\Bigg) \Bigg) \Bigg] . \end{eqnarray*}$$

For the evolution of |1〉〈2| and |2〉〈1|, we use equations (A.1)–(A.3) with ρ₁₂(0) = ρ₂₁(0) = 1 and all other ρ_ij = 0. Having obtained ρ₂₂(t), ρ₃₃(t) follows from equation (A.3). For the evolution of |2〉〈2|, we use equations (A.1)–(A.3) with ρ₂₂(0) = 1 and all other ρ_ij = 0. We thus obtain maps of the form

$$\begin{eqnarray*} &&\fl \left|1\right\rangle \left\langle 2\right| \rightarrow \rho _{11}^{\prime }(t)\left|1\right\rangle \left\langle 1\right|+\rho _{12}^{\prime }(t)\left|1\right\rangle \left\langle 2\right|\nonumber\\ &&+\, \rho _{21}^{\prime }(t)\left|2\right\rangle \left\langle 1\right|+\rho _{22}^{\prime }(t)\left|2\right\rangle \left\langle 2\right|+\rho _{33}^{\prime }(t)\left|3\right\rangle \left\langle 3\right|, \nonumber \end{eqnarray*}$$

$$\begin{eqnarray*} &&\fl \left|2\right\rangle \left\langle 2\right| \rightarrow \rho _{11}^{\prime \prime }(t)\left|1\right\rangle \left\langle 1\right|+\rho _{12}^{\prime \prime }(t)\left|1\right\rangle \left\langle 2\right|\nonumber\\ &&+\, \rho _{21}^{\prime \prime }(t)\left|2\right\rangle \left\langle 1\right|+\rho _{22}^{\prime \prime }(t)\left|2\right\rangle \left\langle 2\right|+\rho _{33}^{\prime \prime }(t)\left|3\right\rangle \left\langle 3\right|, \end{eqnarray*}$$

with ρ'_ij and ρ''_ij determined as already explained.

A.1. Evolution regimes and adiabatic approximation

We can distinguish two markedly different regimes for the evolution of our system. On the one hand, we identify the underdamped regime, in which the cavity sustains a photon long enough for the excitation to come back to the atom. This regime is defined by α² ≡ κ² − 4Ω² < 0. The results of appendix A show that this condition leads to an oscillatory decay of the atom's higher level population. On the other hand, we recognize the purely dissipative regime, in which α² > 0, so that the atom cannot perform Rabi oscillations.

Moreover, and as a subset of the purely dissipative regime, we establish the overdamped regime, characterized by κ ≫ Ω (i.e. α² ≫ 0). In this regime, the probability ρ₂₂ for a photon to be in the cavity decays very rapidly, and so also the coherences ρ₁₂ and ρ₂₁. The cavity remains thus practically all the time in its vacuum state, acting as a fully dissipative environment for the atom. Under these circumstances, we may apply the adiabatic approximation to the evolution of ρ₁₂ + ρ₂₁ (see equation (A.1)), which results in

$$\begin{equation*} \rho _{12}+\rho _{21}\approx \frac{2\Omega }{\kappa }\rho _{11}. \end{equation*}$$

Inserting this expression into the equation for dρ₁₁/dt gives

$$\begin{equation} \frac{{\rm d}\rho _{11}}{{\rm d}t}=-\frac{\Omega ^{2}}{\kappa }\rho _{11}=-\Gamma \rho _{11}. \end{equation} \tag{ A.6 }$$

Hence, in the overdamped regime, ρ₁₁ decays asymptotically with an effective decay constant Γ ≡ Ω²/κ. From the atom's perspective, the cavity behaves as a vacuum-like reservoir.

Taking as the initial state |Ψ_AB(0)〉 = (cos θ|g_A〉|g_B〉 + sin θ|e_A〉|e_B〉)⊗|0_A〉|0_B〉, the initial density matrix is given by

$$\begin{eqnarray*} &&\fl \rho _{AB}(0)=\cos ^{2}\theta |3,3\rangle \langle 3,3|+\cos \theta \sin \theta [|3,3\rangle \langle 1_{,}1|\nonumber\\ &&+\, |1,1\rangle \langle 3_{,}3|] +\sin ^{2}\theta |1,1\rangle \langle 1,1|. \end{eqnarray*}$$

To obtain ρ(t), we apply K_ij(t) = K_(A)i(t)⊗K_(B)j(t) to each |k_Ak_B〉〈l_Al_B|, that is,

$$\begin{eqnarray*} &&\fl K_{ij}(t)|k_{A}k_{B}\rangle \langle l_{A}l_{B}|K_{ij}^{\dagger }(t)\nonumber\\ &&=[ K_{(A)i}(t)|k_{A}\rangle \langle l_{A}|K_{(A)i}^{\dagger }(t) ] \otimes [ K_{(B)j}(t)|k_{B}\rangle \langle l_{B}|K_{(B)j}^{\dagger }(t)]. \end{eqnarray*}$$

The actions of K_(A)i and K_(B)i are formally the same. Hence, it suffices to obtain the map for a single atom–cavity unit. Using the results of appendix A, one finds that ρ_AB(0) evolves to

$$\begin{eqnarray*} &&\fl \rho _{AB}(t)= \cos ^{2}\theta \left|3_{A}\right\rangle \left\langle 3_{A}\right|\otimes \left|3_{B}\right\rangle \left\langle 3_{B}\right|\\ &&+\, \cos \theta \sin \theta ( \rho _{31}|3_{A}\rangle \langle 1_{A}|+\rho _{32}|3_{A}\rangle \langle 2_{A}|) \nonumber\\ &&\otimes ( \rho _{31}|3_{B}\rangle\langle 1_{B}|+\rho _{32}|3_{B}\rangle \langle 2_{B}|) +{\rm h.c}. \end{eqnarray*}$$

$$\begin{eqnarray} &&+\, \sin ^{2}\theta ( \rho _{11}|1_{A}\rangle \langle 1_{A}|+\rho _{12}|1_{A}\rangle \langle 2_{A}|+\rho _{21}|2_{A}\rangle \langle 1_{A}|\nonumber\\ &&+\,\rho _{22}|2_{A}\rangle \langle 2_{A}|+\rho _{33}|3_{A}\rangle \langle 3_{A}|) \otimes ( A\rightarrow B) , \end{eqnarray} \tag{ B.1 }$$

with ρ_ij as defined in appendix A.

Although not relevant for our present purposes, we briefly discuss the alternative non-Markovianity measure that is based on the divisibility of a quantum map Φ. We recall that a completely positive trace preserving (CPTP) map Φ(t + τ, 0) is divisible, if it can be decomposed as the product of two CPTP maps, according to Φ(t + τ, 0) = Φ(t + τ, t)Φ(t, 0). Rivas et al [17] define a map to be Markovian iff it is divisible, which amounts to require that Φ(t + τ, t) is CP [17]. It can be shown that Φ(t + τ, t) is CP iff

$$\begin{equation} g(t)=\lim _{\epsilon \rightarrow 0}\frac{\left|\Phi (t+\epsilon ,t)\otimes I_{d}\left|\alpha \right\rangle \left\langle \alpha \right|\right|-1}{\epsilon }>0, \end{equation} \tag{ C.1 }$$

where |α〉〈α| = d^−1/2∑^d_{i = 1}|i〉|i〉 is a maximally entangled state of the open system—whose dynamics is ruled by Φ—and some ancillary system, both of them having dimension d. When the system's evolution is ruled by a local-in-time master equation ${\rm d}\rho (t)/{\rm d}t=\mathcal {L}(t)\rho (t)$, then for → 0 we can formally write $\Phi (t+\epsilon ,t)=\exp \left[ \mathcal {L}(t)\epsilon \right]$. In this case,

$$\begin{equation} g(t)=\lim _{\epsilon \rightarrow 0}\frac{\left| \left( I_{d}\otimes I_{d}+\epsilon \mathcal {L}(t)\otimes I_{d}\right) \left| \alpha \right\rangle \left\langle \alpha \right| \right| -1}{\epsilon }. \end{equation} \tag{ C.2 }$$

The non-Markovianity measure $\mathcal {N}_{{\rm RHP}} \in \left[ 0,1\right]$, proposed by Rivas et al , is defined as $\mathcal {N}_{{\rm RHP}}= \mathcal {I}/(\mathcal {I}+1)$, with $\mathcal {I}=\int g(t)\,{\rm d}t$. Now, for the kind of approach that we have followed here, $\mathcal {N}_{{\rm RHP}}$ is not directly applicable. The maps we have considered are obtained by tracing out some subsystem of the whole system. On the other hand, the calculation of Φ(t + , t)⊗I_d|α〉〈α| requires that we have Φ(t + , t)|i〉〈j| for any element |i〉〈j| of the system. Let us take, e.g., the map giving the atom–atom evolution ρ_aa(θ, t), equation (4). It was obtained by tracing ρ_AB(t) over the cavities. The space $\mathcal {H}_{aa}$ in which ρ_aa(θ, t) operates is thus spanned by {|gg〉, |ge〉, |eg〉, |ee〉} and the ancilla space required for the calculation of g(t) would be a copy of $\mathcal {H}_{aa}$. In order to calculate g(t), we thus need Φ_aa(t + , t)|gg〉〈gg|, Φ_aa(t + , t)|ge〉〈ge| and so on. Such quantities cannot be unambiguously obtained from the available map Φ(t, 0): ρ_AB(0)↦ρ_AB(t) = Φ(t, 0)ρ_AB(0). Indeed, there are several different initial states ρ_AB(0) which, after tracing over the cavities, lead to the same atom–atom state, say, |gg〉〈gg|. This is of course a consequence, as pointed out by Rivas et al [17], of our having decided to be blind to one part of the whole system by tracing over that part. Nevertheless, if we want to assess—by means of its divisibility—the non-Markovianity of the effective map leading to ρ_aa(θ, t), then we need to restrict the general definition of $\mathcal {N}_{{\rm RHP}}$, as we did with the measure $\mathcal {N}_{{\rm BLP}}$ defined by Breuer et al [14] (denoted by $\mathcal {N}$ in the main text). To this end, we proceed as follows.

The map Φ_AB(t, 0): ρ_AB(0)↦ρ_AB(t) = Φ_AB(t, 0)ρ_AB(0) can be obtained from the two CPTP maps Φ_S(t, 0): ρ_S(0)↦ρ_S(t) = Φ(t, 0)ρ_S(0), S = A, B, that correspond to two open subsystems. Each Φ_S(t, 0) can be expressed in terms of Kraus operators, from which one obtains [16]

$$\begin{equation} \rho _{ii^{\prime }}^{A}(t)=\sum _{ll^{\prime }}A_{ii^{\prime }}^{ll^{\prime }}\rho _{ll^{\prime }}^{A}(0), \ \ \ \rho _{jj^{\prime }}^{B}(t)=\sum _{ll^{\prime }}B_{jj^{\prime }}^{mm^{\prime }}\rho _{mm^{\prime }}^{B}(0). \end{equation} \tag{ C.3 }$$

It then follows that

$$\begin{equation} \rho _{ii^{\prime },jj^{\prime }}^{AB}(t)=\sum _{ll^{\prime }}A_{ii^{\prime }}^{ll^{\prime }}B_{jj^{\prime }}^{mm^{\prime }}\rho _{ll^{\prime },mm^{\prime }}^{AB}(0). \end{equation} \tag{ C.4 }$$

In our case, $A_{ii^{\prime }}^{ll^{\prime }}$ and $B_{jj^{\prime }}^{mm^{\prime }}$coincide with one another, because we have taken two identical atom–cavity subsystems. Furthermore, we obtain $A_{ii^{\prime }}^{ll^{\prime }}$ for → 0 directly from the Liouvillian $\mathcal {L}=-{\rm i}[H,\rho ]+\frac{\kappa }{2} \left( 2a\rho a^{\dagger }-a^{\dagger }a\rho -\rho a^{\dagger }a\right)$, as $A=I+\epsilon \mathcal {L}$. Replacing this in equation (C.4), we obtain the total map Φ(t, 0): ρ_AB(0)↦ρ_AB(t) = Φ_AB(t, 0)ρ_AB(0) to second order in . Tracing ρ_AB(t) over the cavities, we obtain

$$\begin{eqnarray} \fl {\rho _{aa}(t)= \left( \begin{array}{@{}c@{\;\;}c@{\;\;}c@{\;\;}c@{}}( \rho _{22,22}+\rho _{22,33}+\rho _{33,22}+\rho _{33,33}) & ( \rho _{22,31}+\rho _{33,31}) & ( \rho _{31,22}+\rho _{31,33}) & \rho _{31,31} \\ ( \rho _{22,13}+\rho _{33,13}) & ( \rho _{22,11}+\rho _{33,11}) & \rho _{31,13} & \rho _{31,11} \\ ( \rho _{13,22}+\rho _{13,33}) & \rho _{13,31} & ( \rho _{11,22}+\rho _{11,33}) & \rho _{11,31} \\ \rho _{13,13} & \rho _{13,11} & \rho _{11,13} & \rho _{11,11} \end{array} \right),} \nonumber\\ \end{eqnarray} \tag{ C.5 }$$

with explicit expressions for the $\rho _{ll^{\prime },mm^{\prime }}$ that are given to second order in . The map fixing ρ_aa(θ, t), equation (4), is obtained for initial states of the form given by equation (3). We can thus restrict ourselves to the subspace spanned by {|11〉, |33〉} and use an ancillary system that is a replica of this subsystem, so that |α〉 = 2^−1/2(|11〉⊗|11〉 + |33〉⊗|33〉). Using equation (C.4), we obtain Φ_AB(|11〉〈11|), Φ_AB(|11〉〈33|),Φ_AB(|33〉〈11|) andΦ_AB(|33〉〈33|). Tracing these quantities over the cavities yields a matrix of the form of equation (C.5) in each case, for both the system and the ancilla. There is no ambiguity in this case, as both |11〉 and |33〉 correspond to zero cavity photons. With these results, we can calculate g(t) as defined by equation (C.2). It turns out that the effective map, i.e. the result of calculating $\left( I_{d}\otimes I_{d}+\epsilon \mathcal {L}(t)\otimes I_{d}\right) \left| \alpha \right\rangle \left\langle \alpha \right|$ followed by tracing over the cavities, yields a result that differs from the identity I_d⊗I_d only in terms that are of second order in . As a consequence, g(t) = 0 for the evolution of ρ_aa(θ, t), which according to this restricted measure should thus be diagnosed as a Markovian one for all values of the parameters θ, ω, Ω and κ. We thus obtain a diagnosis that occasionally differs from the one based on the trace distance between states. Now, generally, whenever $\mathcal {N}_{{\rm BLP}}(\Phi )>0,$ $\mathcal {N}_{{\rm RHP}}(\Phi )>0$, in contrast to our result. However, we stress that such a result should be taken with a grain of salt, as it does not refer to the actual quantum map Φ(t, 0): ρ_AB(0)↦ρ_AB(t) = Φ(t, 0)ρ_AB(0). If our aim were to compare the two (unrestricted) measures, $\mathcal {N}_{{\rm RHP}}$ and $\mathcal {N} _{{\rm BLP}}$, then we should address the total system AB. However, in such a case, an appropriate entanglement measure like concurrence is lacking, and this precludes addressing the interplay between entanglement and non-Markovianity, which is our main concern here.