In recent decades, there is a great deal of evidence that quantum phenomena play a central role in the development of information theory. Coherent superposition is one of these features which is usually referred as quantum coherence. The non-local quantum correlations among composite subsystems is called entanglement.^[1] The importance of quantum entanglement arises from its various exciting applications such as quantum teleportation,^[2,3] quantum cryptography,^[4] sensitive measurements,^[5] quantum telecloning,^[6,7] and superdense coding.^[8] Due to the rapid growth of the applications of these kinds of quantum states in quantum information processing implementations, a great deal of attention has been paid to the generation and detection of entangled states. Most of these proposals rely on the interaction of atoms (real or artificial) with optical cavities.^[9] Other proposals include quantum dots,^[10] atomic ensembles,^[11] superconducting quantum interference devices,^[12,13] photon pairs,^[14] superconducting qubits,^[15] and trapped ions.^[16–18]

Approximately, all of the introduced schemes depend on the interactions (direct or indirect) between subsystems. For instance, it has been shown that the Jaynes–Cummings model (JCM), which describes the interaction of atoms (two- or multi-level) with cavity field,^[19] could generate entanglement between an atom and a quantized field. This model has been extended to include the interaction of multiple atoms with a multi-mode electromagnetic field.^[20] Thanks to the nonlocality of quantum correlations, it is possible to entangle two or more particles which are distributed over distances without any interactions and, or common history. This phenomenon is called entanglement swapping.^[21,22] In this protocol, the basic recipe is to make a more general system. Then by projection of the quantum state of the whole system onto a maximally entangled Bell state, it is possible to swap the entanglement between these subsystems. There have been many works on this interesting topic. For instance, it has been generated for continuous-variable systems in Ref. [23]. Multiparticle entanglement swapping has been studied in Ref. [24]. The unconditional entanglement swapping has been experimentally demonstrated in Ref. [25]. In Ref. [26], the authors have discussed this phenomenon using a quantum-dot spin system. Researchers have already shown that entanglement swapping could be used for the optimization of entanglement purification.^[27] One-cavity scheme enabling to implement delayed choice for entanglement swapping in cavity QED has been investigated in Ref. [28]. The effect of detuning and Kerr medium on the entanglement swapping has been studied in Ref. [29]. Recently, a high-fidelity unconditional entanglement swapping experiment in a superconducting circuit has been performed in Ref. [30]. Very recently, the swapping of entangled states between two pairs of photons emitted by a single quantum dot has been performed experimentally.^[31]

On the other hand, contrary to the closed systems which are ideal, the real physical systems are open. This means that dissipation is always present in those systems. Actually, the inescapable interaction between the aim system and its surrounding environment makes the entanglement fragile.^[32,33] Because a long-lasting entangled state is an essential resource for the quantum information theory, many strategies have been devoted to fight against the destructive environmental effects: the theory of open quantum systems.^[34–42] However, it should be noted that the idea of interaction of a quantum system with the surrounding environment is not always bad. For instance, it has been shown that there exists a long-living entangled state due to the interaction of a two-qubit system with a common environment.^[43–47] This idea has been generalized to an arbitrary number of qubits inside an environment.^[48–50] Moreover, quantum reservoir engineering has been proven to be useful in stabilizing open quantum systems^[51] and remote entanglement and concentration,^[52] etc. Recently, it has been shown that an external classical field is a practical scheme to preserve the entanglement between two dissipative systems.^[53] In this regard, many studies such as non-Markovianity,^[54] quantum speedup,^[55] quantum Fisher information^[56] have been presented.

Recent experimental schemes in quantum information processing rely on the control of single qubits inside (optical) cavities. However, in practical implementations, achieving a static state of qubits in a cavity is a difficult task, although it is not impossible. In a pioneering work, the effect of the movement of two qubits inside non-Markovian environments on the protection of the initial entanglement has been studied in Ref. [57]. Moreover, other studies illustrated the effect of movement of qubits (both uniform and accelerated) on the interaction between such qubits and electromagnetic radiation.^[58,59] This includes the relativistic velocities for qubits.^[60,61] In a very recent paper, the authors showed the positive role of movement of qubits on the entanglement dynamics of an arbitrary number of qubits in a Markovian and/or non-Markovian environment.^[62] It has also been shown that when all of the qubits have the same velocity, the stationary state of entanglement is independent of the velocity of qubits.^[62]

All of the statements mentioned above motivate us to examine the effect of movement of qubits on the entanglement swapping between two separate subsystems. To end this, we consider two independent cavities, each contains a moving two-level system (qubit) in the presence of dissipation. We model the environment as a set of infinite quantized harmonic oscillators. We take the situation in which each qubit is allowed to move along the cavity axis. We also consider the non-relativistic velocities for qubits. In this situation, the exact dynamics for each subsystem is obtained for both weak and strong coupling regimes corresponding to bad and good cavity limits. We first explore the role of movement of the qubit on the entropy evolution for each subsystem. After that, we perform a Bell state measurement (BSM) on the cavity fields leaving the cavities. This swaps the entanglement between the qubit and the field in each cavity into qubit-qubit and field-field entanglements. We use the concurrence measure^[63] to quantify the amount of swapped entanglement. Naturally, this depends on the initial state of the qubits. Our parametrization for the initial states of qubits allows us to establish an input-independent dynamics of entanglement by taking a statistical average over the initial states of two qubits. This is called entangling power originally introduced for unitary maps^[64] and then generalized for dissipative channels.^[65]

The rest of the paper is organized as follows. In Section 2 the model and the various related parameters are introduced, then the dynamics of each subsystem is obtained. Section 3 deals with the effect of the movement of the qubits on the entropy evolution of each subsystem. We study the entanglement swapping phenomena in detail in Section 4 Finally, in Section 5, we summarize the paper.

We consider two similar, but separate dissipative cavities, each containing a moving two-level atom (qubit) with an excited (ground) |e⟩ (|g⟩) state. These states are separated by transition frequency ω_qb. Each qubit is taken to move along the z-axis of the corresponding cavity with constant velocity v (see Fig. 1). The movement of qubits is characterized by a sine term due to the boundary conditions. The Hamiltonian for each system (with ħ = 1) is given by

Fig. 1.

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	Fig. 1. Pictorial representation of a setup in which a moving qubit is interacting with a dissipative cavity.

Formally, it is more convenient to work in the interaction picture. The Hamiltonian (1) in the interaction picture is given by

We suppose that there is no excitation in the cavities before the occurrence of interaction and each atom is in the coherent superposition of the exited |e⟩ and ground state |g⟩ as

Using the time-dependent Schrödinger equation in the interaction picture, we are ready to have the integro-differential equation for the amplitude ,

As is seen, depends on the spectral density as well as the shape function of the qubit motion. In the continuum limit for the reservoir spectrum, the sum over the modes is replaced by the integral

Substituting (11) into (9), we arrive at the following expression for F(t,t′):

In the above section, we have solved the Schrödinger equation for the case of a moving qubit inside a cavity in both non-dissipative and dissipative regimes. Before we consider the entanglement swapping phenomena, we intend to illustrate the effect of the movement of the qubits on the entanglement dynamics. Among the various measures for computing the degree of entanglement between bipartite systems, we use the linear entropy defined as^[69]

Using Eqs. (7) and (16), the linear entropy at any time can be derived as

Figure 2 illustrates the linear entropy (17) for θ = 0 and the average of linear entropy (20) in the weak coupling regime (R = 0.1) for different values of velocity of the qubit. Taking a glance at these plots reveals the positive role of the movement of the qubit on the survival of the linear entropy. As is seen, by increasing the velocity of the qubit, the linear entropy reaches its maximum value at longer times. The decaying behavior of linear entropy represents a Markovian process. The maximum amount of linear entropy is obtained for θ = 0.

Fig. 2.

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	Fig. 2. Time evolution of (a) linear entropy for θ = 0 and (b) the average of linear entropy versus scaled time τ for weak coupling regime, i.e., R = 0.1 with β = 0 (solid black line), β = 2 × 10−9 (dashed red line) and β = 4 × 10−9 (dot-dashed green line). In these plots we have set ω0/λ = 1.5 × 109.

In Fig. 3 we have plotted the linear entropy for θ = 0 and the average of linear entropy in the strong coupling regime (R = 10) for several motion situations of the qubit. Again, the positive role of the movement of qubit on the survival of the linear entropy is quite clear. Increasing the velocity of the qubit not only makes the linear entropy survive at longer times but also washes out the oscillatory behavior of the linear entropy. These oscillations are a sign of the non-Markovian process. Actually, in this coupling regime, the interaction between the qubit and its surrounding environment is so strong that part of the information that has been taken by the environment is fed back to the qubit. As is clear, in the presence of the movement of the qubit, the sudden death of linear entropy is no longer seen.

Fig. 3.

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	Fig. 3. Time evolution of (a) linear entropy for θ = 0 and (b) the average of linear entropy versus scaled time τ for strong coupling regime, i.e., R = 10 with β = 0 (solid black line), β = 10 × 10−9 (dashed red line) and β = 15 × 10−9 (dot-dashed green line). Again ω0/λ = 1.5 × 109.

In the above section, we examined the positive role of the movement of a single qubit on the entropy evolution of the qubit and its surrounding environment. In this section, we consider two similar but separable systems introduced in Section 2. The time evolution of each system is given in Eq. (7). As is expected, their states are separable, i.e.,

Fig. 4.

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	Fig. 4. Pictorial illustration of the entanglement swapping phenomenon. In each subsystem, the moving qubit is interacting with its own environment. Because of the nonperfect reflecting mirrors of the cavities, the photons can leak out of the cavities. Then a Bell state measurement is performed on these photons to establish the entanglement between the qubits.

Experimentally, there are many qubits inside each cavity. In this way, the qubits inside each dissipative cavity can be considered as a Bose–Einstein condensate in an optical cavity^[70] in the presence of thermal noise which plays the role of the dissipation. Then, the output of the optical cavity of each node is sent to an intermediate site where a Bell-like detection is performed on these optical pairs.^[70]

According to the wave function (7), one observes that there exist only the vacuum and first excited states of the field modes in each system. Therefore, it is possible to consider the following Bell-like states of the fields:^[71]

In the continuation, we consider the following projection operator:

The above relation is fulfilled with the following set of solutions:

θ₁ = θ₂ = 2nπ and arbitrary values of ϕ₁ and ϕ₂ with n = 0,1,2,…;

θ₁ = θ₂ and ϕ₁ − ϕ₂ = 2mπ, with m = 0, ± 1.

These conditions lead to the maximally entangled Bell state (up to an irrelevant global phase)

Again, similar to the above section, we can establish an input-independent dynamics for the swapped entanglement, which is called entangling power. This is carried out by taking a statistical average over all initial states^[65]

Figure 5 illustrates the entangling power as a function of scaled time τ = λ t in the strong coupling regime (i.e., R = 10) for different values of β. In this case, entangling power has an oscillating behavior which is a characteristic feature of non-Markovian regime. Entanglement sudden death phenomenon is clearly seen. As is observed, the entangling power exhibits an oscillatory decay behavior in the absence and presence of the movement of the qubits. However, the presence of movement of qubits makes the decay of entanglement becomes slow. This can be understood by paying attention to the fact that the entanglement between the two qubits depends directly on the entanglement between the qubit and its surrounding filed in each cavity. Therefore, the less entanglement between qubit and field (in each cavity), the less swapped entanglement between qubits. Thanks to the results presented in Section 3, we already know the positive role of the movement of qubit on the entanglement between the qubit and the cavity field. This means that, by choosing a suitable value of β, a long-living stationary entangled state between two qubits can be created.

Fig. 5.

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	Fig. 5. Time evolution of the entangling power of the atom-atom state after BSM () as a function of scaled time τ for strong coupling regime, i.e., R = 10 with β = 0 (solid blue line), β = 10 × 10−9 (dashed red line) and β = 15 × 10−9 (dot-dashed green line). We have set ω0/λ = 1.5 × 109.

In order to explore a deep insight into the role of movement of the qubits on the swapped entanglement, we have shown the density matrix of two qubits at the scaled time τ = 1 for two values of β (see Fig. 6). We have considered the initial state of the qubits to be . As is observed, the population of the state |g,g⟩ (which is zero at the beginning of the interaction) is increased due to the dissipation sources in the cavities. It is evident that by increasing the value of β, the population of this state is decreased, which leads to the survival of swapped entanglement at more significant times.

Fig. 6.

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	Fig. 6. Histogram of density matrix of two qubits at scaled time τ = 1 with initial state for strong coupling regime, i.e., R = 10 with (a) β = 0 and (b) β = 15 × 10−9. In these plots ω0/λ = 1.5 × 109.

In Fig. 7, we have illustrated the entangling power (33) as a function of scaled time τ = λ t in the weak coupling regime for various values of β. Again, the entangling power has decaying behavior. In this case, the oscillatory behavior is no longer observed. The positive role of the movement of the qubits is clearly apparent.

Fig. 7.

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	Fig. 7. Time evolution of the entangling power of the atom-atom state after BSM () as a function of scaled time τ for weak coupling regime, i.e., R = 0.1 with β = 0 (solid blue line), β = 2 × 10−9 (dashed red line) and β = 4 × 10−9 (dot-dashed green line). We have set ω0/λ = 1.5 × 109.

Finally, we take into account the effect of initial qubit state on the evolution of the swapped entanglement. In Fig. 8, we plot the time evolution of the concurrence for different initial states of the qubits. Again we consider both Markovian and non-Markovian environments. It is revealed that both coupling regimes lead to the same stationary value of entanglement. We observe the similar behavior of entanglement for different initial states. However, the amplitude of entanglement depends on the initial state of qubits.

Fig. 8.

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	Fig. 8. Time evolution of the concurrence for different initial states of the qubits for (a) R = 10 and (b) R = 0.1. We have set β = 2 × 10−9, ω0/λ = 1.5 × 109 and , ϕ1 = 0, , ϕ2 = 0 (solid blue line), , ϕ1 = 0, θ2 = 0, ϕ2 = 0 (dashed red line) and , ϕ1 = π, , ϕ2 = 0 (dot-dashed green line).

In this work, we have considered a model to study the possibility of entanglement swapping between two subsystems each contains a moving qubit inside an environment. Our model allows us to treat the environment in both weak and strong coupling regimes corresponding to bad and good cavity limits. By good cavity limit, we have meant an oscillation behavior of entanglement which is due to the memory depth of the environment. We treat the movement of qubits to be classical. Under certain conditions, we have solved the time-dependent Schrödinger equation for each subsystem.

Before considering the entanglement swapping phenomenon, we have examined the influence of the movement of qubits on the entropy evolution of subsystems. This gives us an insight into the possible role of the movement of qubits on the entanglement dynamics. Our parametrization for the initial state of the qubits in each subsystem allows us to construct an initial state independently for the entropy (and later on for the swapped entanglement). This has been carried out by taking a statistical average over all of the initial states of the qubits. The results show the entanglement between qubits and its surrounding environment due to the interaction among them. However, due to the environmental effects, the entropy has decaying behavior. This deterioration of entanglement is suppressed in the presence of movement of the qubit.

Then we turn to the problem of entanglement swapping between such two subsystems. Our goal is to swap the stored qubit-field entanglement in each subsystem into qubit-qubit and field-filed entanglements. Since the cavities are not perfect, the photons can leak out them. This allows us to perform a BSM on the photons leaving the cavities. We have obtained the analytical expression for the normalized state of qubit-qubit after BSM. We have used the concurrence parameter to quantify the amount of swapped entanglement. Several interesting and noticeable points are found. First of all, by considering a suitable Bell state for field modes, the concurrence would be independent of the shape of the incoming photons. Second, there is a set of the initial states which lead to a long-lived maximally entangled state corresponding to stationary state . On the other hand, we have determined the initial states which lead to no entangled state at all. Therefore, the average of entanglement over all initial states of the qubits determines whether our model is a good entangler or not, the notion of entangling power. From the information supplied, the average of swapped entanglement is always greater than zero, which signifies that our proposed model is a good entangler. The positive influence of the movement of qubits on the swapped entanglement is clearly apparent for both weak and strong coupling regimes. Since the velocities of the qubits are treated classically, it is possible to reach the velocities for qubits in which a nearly stationary swapped entanglement is obtained.

The present work can be considered as an extension of Ref. [72] where it has been shown that entanglement swapping in the presence of dissipation can be performed. First of all, by considering a static state of motion for qubits (β = 0), we recover the results presented in the mentioned work. Therefore, this guarantees the validity of our model. Second, it is interesting to notice that the long-lived maximally swapped entanglement (corresponding to a Bell state) (see Eq. (34)) is independent of the movement of qubits. This is exactly the result reported in Ref. [72]. In very recent work, the possibility of entanglement swapping in the presence of dissipation has been studied in Ref. [73] when the qubits are driven by an external classical laser field. Again, it is shown that the long-lived maximally swapped entanglement state can be achieved under certain conditions. Moreover, this state is independent of the effect of the classical field. Altogether, we can conclude the superiority of considering the movement of qubits, especially when the speed of qubits is reasonably large. Under this condition, the swapped entanglement remains static over sufficiently long times, a new result that was not reported in Ref. [72].

We should emphasize that our results could be used in quantum communication applications. The main idea behind these protocols is to transmit and exchange quantum information (entangled states) over long distances.^[74] For instance, generating and swapping entanglement are at the heart of quantum repeater protocols. In this regard, since the environmental effects are always present, our results could boost the efficiency of such protocols. We also note that our results could be useful in the preparation of quantum states. For instance, in order to prepare an entangled state between two qubits that are located in their own distinct cavities, it is enough to detect a photon from a cavity. If we do not know that from which cavity the detected photon is coming, the qubits will be in an entangled state.