1. Introduction

The radiative properties of emitters (e.g. atoms or quantum dots) located inside a dielectric or conducting material can be significantly modified compared to those in vacuum. The modification results from the variation of the density of modes of the EM field which can be adjusted by changing the geometric shape or space period structure of the material [1–3]. The radiative properties of an emitter can also be modified by locating the emitter close to the surface of a dielectric or conducting material [4–9]. In this case, the modification originates from the presence of electromagnetic modes propagating along the surface, called surface plasmon polaritons, or shortly surface plasmons (SP) [10–17]. A new category of materials has been proposed, so-called meta-materials [18–21], characterized by specifically designed geometrical structures which drastically modify the density of the EM modes, so the field propagation, also yielding to the SP field [22–27]. Owing to the high local density of modes, the radiative properties of the emitters can suffer a sharply changed when couple to the surface field [28–34], which has potential applications and attracts a great deal of attention. For example, when quantum dots are placed at a distance about several tens of nanometers above two dimensional metal surface, strong coupling generated between the quantum dots and the collective mode would results in the Rabi oscillation [35]. In the structure composed of zero index and left hand materials, maximum quantum interference and a suppression of the atomic decay rate can be achieved between Zeeman levels [36] due to the anisotropy of the EM modes [37]. It has been shown that by changing the strength of the driving field and adjusting position of the quantum dot, the plasmon modes on the surface of a metal-nanoparticles material can introduce asymmetrical features [38] into the fluorescence spectrum [39].

In this paper, we consider the dynamics of two independent atoms coupled to a SP field generated at the interface of two meta-materials, one of a negative permeability (MN) and the other of a negative permittivity (EN). Our focus is on how the plasmon field induced at the interface between the materials changes the dynamics of the atoms, in particular, the population transfer and entanglement. The mathematical approach we adopt here is based on the Green’s function method [40]. The remarkably simple analytical expressions are derived for the probability amplitudes valid for an arbitrary initial state, arbitrary strengths of the coupling constants of the atoms to the plasmon field, and arbitrary distances between the atoms. We find a number of interesting general results in both strong and weak coupling regimes of the atoms to the SP field. In particular, we find a threshold behavior of the coupling constants which separate the non-Markovian behavior of the system from the Markovian one [41]. The Markovian evolution is usually attributed to a weak coupling of an atom to the field. We show that the collective effects may result in the Markovian evolution even in the limit of a strong coupling of the atoms to the field. Inversely, a non-Markovian evolution can be seen even in the regime of a weak coupling.

The system which we discuss here is closely related to systems, for example, involving photonic crystal waveguides [42–45], nanorod resonators [46], nano-wires [47], metal nanoparticles [48], a thin membrane made of a negative left handed material [49], and a quantum dotgraphene structure [50]. In these systems the coupling between emitters is also mediated by surface polaritons, but the calculations are not specifically oriented towards studying the strong coupling effects. The calculations are limited to the weak coupling regime resulting in a Markovian dynamics of the emitters. In this case, the incoherent spontaneous exchange of photons may occur resulting in collective damping of the emitters [51, 52]. The spontaneous exchange of the excitation can lead to entanglement between the emitters which may exist over distances much larger than the resonant wavelength. The study of entanglement between distant atoms and controlled the transmission of information between them are vital to the development of quantum information technology [53, 54].

The plan of this paper is as follows. In Sec. 2 we introduce the model and present the explicit analytic expressions for the time dependence of the probability amplitudes of the symmetric and antisymmetric combinations of the probability amplitudes of the atoms. We assume that a single excitation is present initially in the system and demonstrate how the evolution of the system can be simply understood in terms of the evolution of the atoms and their corresponding images. We then demonstrate in Sec. 3 the collective behavior of the atoms in both Markovian and non-Markovian regimes of the evolution. Entangled properties of the atoms are discussed in Sec. 4, where we calculate the concurrence for different initial states and different coupling strengths of the atoms to the plasmon field. The effect of an off-resonant coupling of the atoms to the plasmon field on the collective dynamics and entanglement is discussed in Sec. 5. We summarize our results in Sec. 6. The paper concludes with an Appendix in which we give details of the derivation of the integro-differential equations for the probability amplitudes and the calculations of the integral kernels. Both longitudinal and transverse parts of the Green function are considered in the evaluation of the kernels.

2. Atoms interacting with the plasmon field

Perfectly conducting materials are known to generate a strong SP field which is refined in a short regime near the surface [55–57]. The plasmon field can also be produced at the surface of a meta-material with either negative permittivity (ε < 0) or negative permeability (μ < 0). However, the density of the SP field only originates from one or several discrete modes, which can be derived by applying the continuous conditions on the boundaries. Recently, Tan et al. [58] have shown that the density of the EM modes can be significantly enhanced at the interface between two meta-materials, one with negative ε and the other with negative μ. Especially, when the materials are perfectly paired, i.e. ε₁ = −ε₂ and μ₁ = −μ₂, the effective permittivity and permeability, defined as

In this paper, we consider a system composed of two identical atoms located at a distance z₀ from the interface between two different negative index material slabs, μ-negative (MN) and ε-negative (EN) slabs, as shown in Fig. 1. For a potential experimental realization of a paired zero-index materials one could consider a structure fabricated by the composite right/left-handed transmission line, in which both the effective permittivity and permeability are zero in the microwave region (ω ~ 2.0 GHz) [61]. An impedance-matched zero index material of resonant frequencies ranging from optical to near infrared region (ω ~ 200 THz), can also be realized by using purely dielectric high index rods or cut wire pairs [62, 63].

We assume that the atoms are located in the MN slab and the distance between the atoms is large so the direct interaction between the atoms is weak and can be ignored, especially when compared to the coupling strength with the plasmon field. Each atom is represented by its ground state |g_l〉 and excited state |e_l〉 (l = 1, 2), the atomic transition frequency ω_a, and the atomic transition dipole moment ${\overrightarrow{p}}_l$. The atoms interact with an electromagnetic field via a dipole interaction according to the Hamiltonian [64]

 

Fig. 1 (Color online) Schematic illustration of the system showing the electric field (yellow arrows) of an electromagnetic wave and dislocated surface charges at the interface between MN and EN slabs. The z axis is taken normal to the interface with its origin at the interface. The slabs have thickness d₁ and d₂, respectively, and are assumed to have infinite extents in the xy plane. Two atoms are embedded in the MN slab at fixed positions (x₁, 0, z₀) and (x₂, 0, z₀), where z₀ is the distance of the atoms from the interface between the materials. The atomic transition dipole moments p₁ and p₂ are parallel to each other and oriented in the x − z plane.

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For the permittivity and permeability of the slabs, we assume that ε₁ and μ₂ are positive constants, but ε₂ and μ₁ are negative and strongly depend on frequency of electromagnetic field, and are given by the following expressions [59, 65]

If the field was initially at t = 0 in the vacuum state and the atoms shared a single excitation, the wave function of the system at time t > 0, written in the interaction picture is of the form

To study the dynamics of the atoms, we consider the wave function of the system whose the time evolution is governed by the Schrödinger equation. We may introduce symmetric and antisymmetric combinations of the probability amplitudes, $C_s=\left(C_1+C_2\right)/\sqrt{2}$ and $C_a=\left(C_1-C_2\right)/\sqrt{2}$, corresponding to collective symmetric and antisymmetric states of the two-atom system. A straightforward but lengthly calculation (for details see Appendix) leads to the following equations of motion

Here, γ ≡ γ_e = γ_m represents Ohmic losses in the materials, δ = ω_s − ω_a is the detuning of the atomic transition frequency from the plasmon field frequency, ${\mathrm{\Omega }}_s^2={\mathrm{\Omega }}_0^2\left[1+U\left(x_{21},z_0\right)\right]$ and ${\mathrm{\Omega }}_a^2={\mathrm{\Omega }}_0^2\left[1-U\left(x_{21},z_0\right)\right]$ are coupling strengths (Rabi frequencies) of the symmetric and antisymmetric states to the plasmon field in which

The function U(x₂₁, z₀) depends on the separation between the atoms, x₂₁, and also their distance z₀ from the interface. It determines the strength of the coupling between the atoms. In the limit of x₂₁ ≫ λ_s, U(x₂₁, z₀) ≈ 0, while for x₂₁ ≪ λ_s, U(x₂₁, z₀) ≈ 1. Thus, for large x₂₁, the effects of the coupling between the atoms become negligible and the atoms evolve independently. Notice that at x₂₁ = 0 the function U(x₂₁, z₀) is always unity independent of the value of z₀, the distance of the atoms from the interface. However, the variation of U(x₂₁, z₀) with x₂₁ depends strongly on z₀. This is illustrated in Fig. 2, which shows U(x₂₁, z₀) as a function of x₂₁ for several different values of z₀. It is seen that for large z₀, the function U(x₂₁, z₀) varies slowly with x₂₁. In that case, the indirect coupling between the atoms which is provided by the plasmon field is effectively quite strong even at large separations. On the other hand, for small z₀, (z₀ ≪ λ_s), the function U(x₂₁, z₀) is different from zero only over very small distances x₂₁ and decays rapidly to zero as x₂₁ increases. Thus, a strong coupling of the atoms to the plasmon field destroys the collective behavior of the atoms. In the physical terms, the location of the atoms at a small distance z₀ from the interface leads to a strong spatial confinement (localization) of the surface plasmon fields around x₁ and x₂ resulting in a weak overlap of the surface fields excited by the atoms.

 

Fig. 2 Dependence of U(x₂₁, z₀) on separation between the atoms x₂₁/λ_s for several different distances of the atoms from the interface: z₀ = 0.05λ_s (solid black line), z₀ = 0.1λ_s (dashed red line), z₀ = 0.25λ_s (dashed-dotted blue line), and z₀ = 0.5λ_s (solid green line).

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From Eq. (9) it follows that the kernel K_j (t, t′) is a function only of the time difference t – t′. Therefore, Eq. (8) can be solved exactly by Laplace transformation, and the solution is

The result (12) shows a number of interesting features. Firstly, the presence of the Ohmic losses γ in the materials introduces a threshold effect that depending upon Ω_j < u or Ω_j > u, the effective Rabi frequencies ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ can be either purely real or purely imaginary. In the other words, the Rabi frequencies may contribute to either damping of the amplitudes or sinusoidal oscillations. Secondly, in the strong coupling limit of Ω_j > u, the losses present in the system are only those of the Ohmic losses γ. Thirdly, below threshold each of the amplitudes C_s (t) and C_a(t) is damped with two different rates, a reduced (subradiant) rate, $\frac{1}{4}\gamma -{\tilde{\mathrm{\Omega }}}_j$, and an enhanced (superradiant) rate, $\frac{1}{4}\gamma +{\tilde{\mathrm{\Omega }}}_j$. The involvement of two rather than a single damping rate seems to contradicts our expectation since, according to Dicke [69] (see also Refs. [70–72]), each of the collective amplitudes of a two atom system should decay with a single rate only. The amplitude C_s (t) should decay with the enhanced rate while C_a(t) should decay with the reduced rate.

 

Fig. 3 (Color online) Two atoms located at a distance z₀ from the interface between two materials and their images located at a distance z₀ behind the interface. The atoms are not directly coupled to each other, but can be coupled by the radiation reflected from the interface. A photon emitted by atom 1 and reflected from the interface towards atom 2 can be viewed as being emitted from the image of the atom 1.

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A qualitative understanding of the involvement of two decay rates in the evolution of both C_s(t) and C_a(t) may be obtained by considering the interaction of the atoms with the plasmon field as the interaction with images of the atoms located at a distance z₀ behind the interface. As illustrated in Fig. 3, the radiation field emitted by either atom 1 or atom 2 and reflected from the interface in the direction normal to the interface can be regarded as the radiation from an image located at a distance z₀ behind the interface.

If we introduce the notation ${\tilde{C}}_j\left(t\right)=C_j\left(t\right)\exp \left[\left(\frac{1}{2}\gamma +i\delta \right)t\right]$ and ${\tilde{C}}_{jI}\left(t\right)=i{\mathrm{\Omega }}_0{\displaystyle {\int }_0^{t}dt\prime {\tilde{C}}_j\left(t\prime \right)}$, where ${\tilde{C}}_{jI}\left(t\right)$ can be interpreted as the probability amplitude of the image j, then it is straightforward to write Eq. (12) in the form

The reason for the involvement of two decay rates in Eq. (12) is now clear: The term decaying with the enhanced rate is associated with the decay of the symmetric superpositions involving the atomic and image states, ${\tilde{D}}_s\left(t\right)$ and ${\tilde{G}}_s\left(t\right)$, whereas the term decaying with the reduced rate is associated with the decay of the antisymmetric superpositions ${\tilde{D}}_a\left(t\right)$ and ${\tilde{G}}_a\left(t\right)$. The slowest decay rate in the system, ${\gamma }_a^{(-)}=\frac{1}{4}\gamma -{\tilde{\mathrm{\Omega }}}_a$, is the decay rate of the antisymmetric superposition ${\tilde{G}}_a\left(t\right)$ whereas the fastest decay rate, ${\gamma }_s^{(+)}=\frac{1}{4}\gamma +{\tilde{\mathrm{\Omega }}}_s$, is the decay rate of the symmetric superposition ${\tilde{D}}_s\left(t\right)$.

We may conclude that the interaction of the atoms with the surface plasmon field can be viewed as the interaction between the atoms and their corresponding images. This also shows that the dynamics of a system composed of two atoms coupled to a plasmon field is analogous to the dynamics of a four-atom system in a rectangular configuration. Thus, one can obtain detailed and exact solutions for the dynamics of four interacting atoms in a rectangular configuration by considering the dynamics of the simpler system, two atoms coupled to a SP field.

3. Markovian and non-Markovian regimes of the evolutions

We have already seen that the Rabi frequencies of the collective states are altered by the interaction U(x₂₁, z₀), with Ω_s enhanced and Ω_a reduced by U(x₂₁, z₀). This results in two effective Rabi frequencies ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ which exhibit a threshold effect that depending upon Ω_j < u, (b) Ω_j > u, the effective Rabi frequencies can be purely real or imaginary. Since Ω_s ≠ Ω_a, the threshold effects occur at different u. Therefore, we can distinguish between three regions of Ω_s and Ω_a: (a) Ω_s < u and Ω_a < u, (b) Ω_s > u and Ω_a < u, (c) Ω_s > u and Ω_a > u. Physically, the threshold values of ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ separate what we can identify as the non-Markovian (memory preserved) regime from the Markovian (memoryless) regime of the evolution [73–76]. In the case (a), in which ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ are real, the Rabi frequencies contribute to the decay rates, which is a manifestation of a Markovian evolution. In the case (b), the dynamics of the system are partly Markovian and partly non-Markovian. The symmetric state undergoes a non-Markovian whereas the antisymmetric state undergoes a Markovian evolutions. In the case (c), the dynamics of the system are fully non-Markovian that the amplitudes of both symmetric and antisymmetric states undergo an oscillatory evolution, which is a manifestation of a non-Markovian evolution. A non-Markovian evolution is a reversible (coherent) process characterized by a continuous oscillation of an initial excitation between the atoms and the field. On the other hand, the Markovian evolution is a irreversible process of a flow of the excitation to the field resulting in the dissipation of an initial excitation. In experimental practice the non-Markovian evolution could be realized by locating the atoms at distances z₀ from the interface such that the resulting magnitudes of Ω_s and Ω_a would be large enough to overcome the losses in the materials determined by the parameter u. The Markovian type evolution, on the other hand, could be achieved by locating the atoms at distances z₀ such that the resulting magnitudes of Ω_s and Ω_a would be smaller than the losses in the materials.

With the experimentally realistic parameters, for example, of the impedance matched structure fabricated by the transmission line [61], where the surface plasmon resonant frequency ω_s ~ 1.8 GHz and the Ohmic losses of the slabs are of a value γ ~ 0.3 MHz, and by taking a typical value for the atomic damping rate Γ_A ~ 4 × 10⁻² MHz, one can easily verify using Eq. (10) that the Rabi frequencies Ω_s and Ω_a can be larger than γ/4 at distances from the interface z₀ < 0.45λ_s.

3.1. Both Ω_s and Ω_a below threshold

Let us specialize Eq. (12) to the case of exact resonance, δ = 0, and first examine the situation when the coupling of the atoms to the plasmon field is weak, Ω₀ ≪ γ. In this case, both ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ could be below threshold. If Ω_j < u, then ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ are real and we see from Eq. (12) that the Rabi frequencies contribute to the damping rates of the probability amplitudes. As we have already explained, this is the kind of behavior corresponding to a Markovian evolution. Since Ω_s ≠ Ω_a, we see that the weak coupling of the atoms to the plasmon field may result in the decay of the probability amplitudes C₁(t) and C₂(t) with four rates, two enhanced and two reduced rates.

Figure 4 shows the time evolution of the populations P₁(t) = |C₁(t)|² and P₂(t) = |C₂(t)|² for both ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ below threshold. At early times, the population P₁ (t) decreases whereas P₂(t) increases until the populations become equal. At that time, the populations began to decay monotonically. The rate they decay is equal to ${\gamma }_a^{(-)}$, the slowest decay rate of the antisymmetric state. Since the atoms are very strongly coupled to each other, U(x₂₁, z₀) = 0.95, the decay rate ${\gamma }_a^{(-)}\approx 0.002\gamma$. Consequently, the effective decay time of the populations can be very long. The decay of the populations is irreversible so the evolution of the system is Markovian.

 

Fig. 4 Time evolution of the populations P₁(t) = |C₁(t)|² (black solid line) and P₂(t) = |C₂(t)|² (red dashed line) for δ = 0, Ω₀ = 0.15γ, and U(x₂₁, z₀) = 0.95, corresponding to both Ω_s and Ω_a below the threshold of 0.25γ, Ω_s = 0.21γ and Ω_a = 0.033γ. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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3.2. Ω_s above threshold and Ω_a below threshold

At large values of U(x₂₁, z₀) it may happen that Ω_a < u and simultaneously Ω_s > u even if the atoms are weakly coupled to the plasmon field, i.e. Ω₀ < u. For example, when U(x₂₁, z₀) ≈ 1, we have Ω_a ≈ 0 and Ω_s ≈ 2Ω₀. Hence, Ω_s can be larger than u even if Ω₀ < u and at the same time Ω_a can be smaller than u. For Ω_s > u the Rabi frequency ${\tilde{\mathrm{\Omega }}}_s$ is purely imaginary, and then the time evolution of the probability amplitude C_s(t) takes the form

Figure 5 shows the evolution of the populations for Ω_s above and Ω_a below the threshold of γ/4. At early times, the oscillations of the populations with the Rabi frequency of the symmetric mode are clearly visible. Meanwhile, the populations oscillate with the Rabi frequency of the symmetric mode. In other words, the evolution of the populations is reversible but the reversibility occurs in a restricted time range $t<1/{\gamma }_a^{(-)}$. Beyond $t~1/{\gamma }_a^{(-)}$ the populations decay monotonically that the evolution is irreversible. Thus, we can clearly distinguish between the non-Markovian and Markovian regimes of the evolutions. We see that the upper limit on time of the reversible evolution results from the presence of the interaction between the atoms. Clearly, it is a collective effect. Physically, it is a consequence of the fact that a large part of the population is trapped in the asymmetric state determined by the amplitude ${\tilde{G}}_a\left(t\right)$ thereby lowering the strength of the coupling of the atoms to the plasmon field. It is easy to see, since for small distances between the atoms U(x₂₁, z₀) ≈ 1, we have Ω_a ≈ 0, which means that the antisymmetric states decouple from the plasmon field. This example also shows that the atoms when behaving collectively can be weakly coupled to the field even if individually they are strongly coupled to the field.

 

Fig. 5 Time evolution of the populations P₁(t) = |C₁(t)|² (black solid line) and P₂(t) = |C₂(t)|² (red dashed line) for δ = 0, Ω₀ = 0.5γ, and U(x₂₁, z₀) = 0.99 corresponding to Ω_s above and Ω_a below the threshold of 0.25γ, Ω_s = 0.705γ and Ω_a = 0.05γ. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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3.3. Both Ω_s and Ω_a above threshold

Above the thresholds, ${\tilde{\mathrm{\Omega }}}_s$ and ${\tilde{\mathrm{\Omega }}}_a$ are purely imaginary. The time evolution of the probability amplitudes is then given by

Figure 6 shows the time evolution of the populations P₁(t) = |C₁(t)|² and P₂(t) = |C₂(t)|² when the atoms are strongly coupled to the plasmon field with z = 0.05λ_s, δ = 0, but are weakly coupled to each other, U(x₂₁, z₀) = 0.1. Note the presence of two characteristic time scales of the oscillations associated with the presence of two slightly different Rabi frequencies. At short times, $t\ll 1/{\overline{\mathrm{\Omega }}}_a$, the initially excited atom 1 periodically exchange the excitation with the plasmon field at the Rabi frequency ${\overline{\mathrm{\Omega }}}_s$. The population of the atom 2 builds up with the oscillation of frequency ${\overline{\mathrm{\Omega }}}_s$. The amplitudes of the populations are modulated with frequency ${\overline{\mathrm{\Omega }}}_a$ causing collapses and revivals of the atomic populations.

 

Fig. 6 Time evolution of the populations (a) P₁(t) = |C₁(t)|² and (b) P₂(t) = |C₂(t)|² for δ = 0, Ω₀ = 25γ, and U(x₂₁, z₀) = 0.1. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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When the atoms are close to each other the collapses and revivals of the populations are absent. Instead, a periodic localization of the excitation is observed. This is illustrated in Fig. 7, which shows the evolution of the populations for a small distance between the atoms at which U(x₂₁, z₀) = 0.8. The manner the populations oscillate is different for P₁(t) and P₂(t). We see a periodic localization of the excitation that even at long times the memory effects are still evident. The explanation of this feature follows from the observation that at small distances between the atoms, the time scale of the oscillations of the antisymmetric state is very large, approaches infinity when U(x₂₁, z₀) → 1. Therefore, the system effectively evolves with a single time scale determined by the Rabi frequency of the symmetric state, t ~ 1/Ω_s.

 

Fig. 7 Time evolution of the populations (a) P₁(t) = |C₁(t)|² and (b) P₂(t) = |C₂(t)|² for δ = 0, Ω₀ = 25γ, and U(x₂₁, z₀) = 0.8. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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4. Evolution of entanglement between the atoms

Given the time evolution of the probability amplitudes, we now proceed to evaluate the concurrence, a measure of entanglement between two qubits [71, 72]. Following the definition of the concurrence, we find that in terms of the amplitudes of the symmetric and antisymmetric combinations, the concurrence is given by the following expression

A positive value of the concurrence, C(t) > 0, indicates entanglement between the atoms, and C(t) = 1 corresponds to maximally entangled atoms. It is clear from Eq. (18) that the atoms are entangled whenever C_s(t) ≠ C_a(t). Otherwise, the atoms are separable. Thus, to examine the occurrence of entanglement between the atoms we must look at differences in the evolution of the amplitudes C_s(t) and C_a(t). If initially, C_s(0) = C_a(0), then according to the solutions of Eq. (12), the amplitudes will evolve differently only if U(x₂₁, z₀) ≠ 0. It then follows that the coupling between the atoms through the plasmon field is necessary to create entanglement between the atoms from an initial separable state.

The features of the concurrence for the three regions of Ω_s and Ω_a are illustrated in Figs. 8–11. Figure 8 shows the evolution of the concurrence in the case of a strong coupling of the atoms to the plasmon field at which Ω_s and Ω_a are above their thresholds and two different values of the interaction strength between the atoms, U(x₂₁, z₀) = 0.1 and U(x₂₁, z₀) = 0.8. The interaction creates a small difference between the frequencies of the oscillation of the symmetric and antisymmetric modes that ${\overline{\mathrm{\Omega }}}_s\ne {\overline{\mathrm{\Omega }}}_a$. The frequency difference induces beating oscillations of the populations of the atoms, as was seen in Fig. 6, and one can see from Fig. 8 that these beating oscillations are rendered visible as beats in the concurrence.

 

Fig. 8 Concurrence versus time for the case of above threshold with Ω₀ = 25γ, δ = 0, (a) U(x₂₁, z₀) = 0.1 and (b) U(x₂₁, z₀) = 0.8. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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Fig. 9 Concurrence versus time for Ω₀ = γ, δ = 0 and U(x₂₁, z₀) ≈ 1 corresponding to the case of Ω_s above threshold but Ω_a below the threshold. Frame (a) shows the concurrence for U(x₂₁, z₀) = 0.95 (solid black line) and U(x₂₁, z₀) = 0.99 (dashed red line). The atoms were initially in a separable state |Ψ(0)〉 = |e₁〉 |g₂〉. Frame (b) shows the concurrence for U(x₂₁, z₀) = 0.95 and two different initial states, the maximally entangled symmetric state |Ψ(0)〉 = |s〉 (solid black line) and the maximally entangled antisymmetric state |Ψ(0)〉 = |a〉 (dashed red line).

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Fig. 10 The time evolution of the populations of the atoms for the situation presented in Fig. 9. Frame (a) shows the populations P₁(t)(solid black line) and P₂(t) (dashed red line) for Ω₀ = γ, δ = 0 and U(x₂₁, z₀) = 0.95. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉. Frame (b) shows the time evolution of the population P₁(t) for two different initial states, |Ψ(0)〉 = |s〉 (solid black line) and |Ψ(0)〉 = |a〉 (dashed green line). Not shown is P₂(t) since in this case P₂(t) = P₁(t).

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Fig. 11 Concurrence versus time for the case of below threshold and two different initial states (a) |Ψ(0)〉 = |e₁〉 |g₂〉 and (b) |Ψ(0)〉 = |s〉 with Ω₀ = 0.15γ, δ = 0 and different U(x₂₁, z₀): U(x₂₁, z₀) = 0.99 (solid black line), U(x₂₁, z₀) = 0.5 (dashed red line), U(x₂₁, z₀) = 0.25 (dashed-dotted blue line).

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Interesting features of the entanglement also appear when the symmetric mode evolves at Rabi frequency which is above the threshold, Ω_s > u, and simultaneously the antisymmetric mode evolves at Rabi frequency which is below the threshold, Ω_a < u. Under this circumstance, the probability amplitude of the symmetric mode is determined by Eq. (17) whereas the amplitude of the antisymmetric mode is given by Eq. (12).

Figure 9 shows the evolution of the concurrence for this special case. We see that the concurrence is zero only at the initial time t = 0. As time progresses the concurrence develops to a nonzero value. The concurrence never becomes zero as time develops, and thus no periodic entanglement quenching occurs. This feature is associated with the fact that with the Rabi frequency Ω_a < u, the population of the antisymmetric state does not evolve in time leading to a trapping of a part of the atomic populations in their energy states. This is illustrated in Fig. 10, which shows the time evolution of the population P₁(t) and P₂(t) for the same parameters as in Fig. 9. We see from Fig. 10(a) that the initial population is periodically transferred between the atoms. However, the transfer is not complete that the populations of the atoms never become zero during the evolution. A part of the population is trapped in the atoms and is not transferred between them. Figure 10(b) shows the evolution of the population P₁(t) for two initial maximally entangled states, |s〉 and |a〉. Since in this case P₂(t) = P₁(t), we clearly see that the concurrence, if starts from maximally entangled state, it follows the evolution of the population of the atoms.

We now turn to the case of a weak coupling of the atoms to the plasmon field. In this case, the interaction of the atoms with the field contributes to the dissipation of the excitation. Figure 11 shows the effect of increasing interaction strength between the atoms on the concurrence for a weak coupling of the atoms to the plasmon field, both Ω_s and Ω_a below threshold, which corresponds to a Markovian evolution of the system. In Fig. 11(a) the system starts from the separable state |e₁〉 |g₂〉, whereas in Fig. 11(b) the initial state of the system is the maximally entangled state |s〉. We see that even in the weak coupling regime, a large and long living entanglement can be created between the atoms. The entanglement created from the initial separable state increases with an increasing U(x₂₁, z₀) and attains the maximal value of C = 0.5 for U(x₂₁, z₀) ≈ 1. The behavior of the concurrence is similar to that noted in the decay of two atoms into a common Markovian reservoir [72].

When the system starts from a maximally entangled state, either |s〉 or |a〉, the initial entanglement always decays to zero with no entanglement present at long times, as illustrated in Fig. 11(b). This is readily understood if it is recalled that the symmetric and antisymmetric states evolve independently in time. Thus, if the population of the antisymmetric state was initially zero it will remain zero for all times. In this case, the system of two atoms effectively behave as a single two-level system with the upper state |s〉 and the ground state |g〉. Then, the initial population of the state |s〉 decays exponentially to the ground state with the rate $2{\mathrm{\Omega }}_s^2/\gamma$.

5. Off-resonant coupling

To this end we have discussed the collective effects induced by the resonant interaction of the atoms with the plasmon field. We have established the importance of the images of the atoms in the atomic dynamics. Moreover, we have demonstrated the equivalence of the system with that of four interacting atoms. We now turn to the off-resonant case of the atomic transition frequencies strongly detuned from the plasmon frequency, δ ≫ γ, Ω_s. Under such condition, the effective Rabi frequency ${\tilde{\mathrm{\Omega }}}_s$ can be approximated by

Thus, if in Eq. (12) the effective Rabi frequency is replaced by (19), we get, up to terms of order ${\mathrm{\Omega }}_s^2/{\delta }^2$,

A similar expression with s → a gives C_a(t). We see that C_s(t) is composed of fast and slow oscillating terms varying in time with frequencies δ and ${\mathrm{\Omega }}_s^2/\delta$, respectively. Of the two terms it is the one of the small magnitude $({\mathrm{\Omega }}_s^2/4{\delta }^2)$ arising from the presence of the images. Thus, the evolution of C_s(t) is well determined without much contribution of the images. It is particularly well seen from Eq. (15) that in the limit of δ ≫ Ω_s, cos ϕ ≈ 1 (sin ϕ ≈ 0) so that the superposition amplitude ${\tilde{D}}_s(t)=0$ and ${\tilde{D}}_a(t)$ is reduced to the atomic amplitude ${\tilde{C}}_s(t)$. In other words, two atoms significantly detuned from the plasmon field are coupled each other by exchanging virtual photons through a short interaction time with the plasmon field.

The above considerations are illustrated in Fig. 12, which shows the time evolution of the atomic populations for δ = 50γ. We see that the atoms exchange the population with frequency $2{\mathrm{\Omega }}_0^2/\delta$. The fast oscillations seen in the early time of the evolution occur at frequency δ and can be attributed to the involvement of the images in the dynamics of the system. The presence of the fast oscillations only at very early times of the evolution is also consistent with energy-time uncertainty arguments. It is easy to understand. At short times the uncertainty of the energy of the atoms and the plasmon field is very large, so one cannot distinguish between ω_a and ω_s. This results in the presence of the fast oscillation of frequency δ. As time progresses, the frequencies become more distinguishable resulting in the disappearance of the fast oscillations.

 

Fig. 12 Time evolution of the populations (a) P₁(t) = |C₁(t)|² and (b) P₂(t) = |C₂(t)|² for δ = 50γ, Ω₀ = 25γ, and U(x₂₁, z₀) = 0.1. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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It is interesting to contrast the entanglement created at δ ≠ 0 with that created in the resonant case of δ = 0. We have seen in Sec. 4 that in the resonant case the maximal entanglement which can be created between the atoms from an initial separable state cannot exceed C = 0.5. For off-resonant case (δ ≠ 0), however, the entanglement can be significantly enhanced. This is shown in Fig. 13, which illustrates the time evolution of the concurrence for a large detuning δ. Clearly, at early times of the evolutions the concurrence is larger than 1/2, increases with an increasing U(x₂₁, z₀) and becoming as large as C = 1. This behavior can be explained in terms of the energy-time uncertainty relation. At short times a large uncertainty in the energy results in a large uncertainty in the localization of the excitation. We see that the increased possibility to distinguish between the frequencies of the atoms and the plasmon field results in an enhanced entanglement between the atoms.

 

Fig. 13 Concurrence versus time for the case of above threshold with Ω₀ = 25γ, δ = 50γ and different U(x₂₁, z₀): (a) U(x₂₁, z₀) = 0.1 and (b) U(x₂₁, z₀) = 0.95. The atoms were initially in the state |Ψ(0)〉 = |e₁〉 |g₂〉.

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6. Conclusions

We have studied the dynamics of two two-level atoms located near to the interface of two meta-materials: one of negative permeability and the other of negative permittivity. We have derived analytical expressions for the probability amplitudes of the atomic states valid for an arbitrary initial state, arbitrary strengths of the coupling constants of the atoms to the plasmon field, and arbitrary distances between the atoms. We have shown that the effect of the surface plasmon field is to produce several interesting features, such as (1), a threshold behavior of the Rabi frequencies of the atoms to the plasmon field which distinguishes between the non-Markovian and Markovian regimes of the evolutions. It has been found that the threshold behavior may lead to three different regimes of the evolution; fully Markovian, simultaneous Markovian and non-Markovian, and fully non-Markovian evolutions. The three regimes determines three different time scales of the evolution of the memory effects and entanglement. (2) In the case of the resonant coupling of the surface plasmon field to the atoms, the field does not appear as a common reservoir to the atoms. We have adopted the image method and showed that in the resonant case the dynamics of the two atoms are analogous to those of the dynamics of a system of four atoms in a rectangular configuration. (3) In the limit of a strong detuning of the plasmon field frequency from the atomic transition frequencies the dynamics resemble those of two atoms coupled to a common reservoir.