Introduction

Two-level system (qubit) is not only the key element in various fields of the modern physics, such as quantum optics and collision physics1,2, but also the fundamental building block of modern applications ranging from quantum control3 to quantum processing4.

Due to the rapid development of experiments in macroscopic solid state physics, the artificial two-level atoms qubits based on the superconducting (SC) circuits5,6 and quantum dots (QDs)7 have been recognized as possible candidate for quantum processing. The SC-qubits have macroscopic quantum coherence. It may be helpful for the realization of the conditional two-qubit gate and quantum hybrid system8,9. Embedding QD-qubits in microcavities enhances the light extraction efficiency via the Purcell effect and permits the study of cavity QED effects in solid-state systems10,11,12. Experimentally, the qubit-photon interaction was intensively investigated8,9,13,14,15,16.

The dissipation eradicates the useful quantum coherence and correlations17,18. The existence of dissipative qubits, such as in amorphous solids, is a longstanding problem in solid-state physics19,20.

There is a growing interest in the dynamics of non-local correlations (NLCs) beyond the quantum entanglement (QE)21, which is a unique type that has a major role in quantum processing22. However, QE does not have all of the non-classical properties of the quantum correlations23. While NLCs between the parts of a system in a pure state is fully characterized by their entanglement, mixed states may possess NLCs even if they are not entangled. New types of NLCs were introduced beyond QE24 as: measurement-induced disturbance25, quantum discord26, and that was determined by using p-norms such as; the Hilbert-Schmidt norm, Schatten one-norm and Bures norm. Due to analytic difficulty to the quantum discord, the geometric correlations appeared via geometric quantum discord (GQD) and the measurement-induced nonlocality (MIN)27 were proposed by using the 2-norm28. However, these measurements that are based on 2-norm have been proved to be incompetent measures of NLC29. Consequently, the GQD and MIN are derived using 1-norm (trace norm)30,31,32. Moreover, non-classical correlations registered by Bell inequality violation33 (that constitutes one of the most striking phenomena ever observed in nature) is used as an indicator of non-local quantum properties.

While NLCs between the parts of a system in a pure state is fully characterized by their entanglement, mixed states may possess NLCs even if they are not entangled. Unlike entanglement, quantum discord is rather robust against decoherence34. It is shown that the dynamics of GQD is more robust than the thermal entanglement. Also, the measures of MINs can be considered as one type of quantum correlation that differs from entanglement and quantum discord30,31,35.

Despite the complexity of the suggested model, it is significant to introduce an analytical description of two optically driven qubits inside an open cavity, that is initially prepared in a coherent state. Therefore, the non-locality correlations [including, trace norm measurement induced non-locality, maximal Bell-correlation] and the entanglement via the concurrence could be determined.

In Sec. 2, the model of (two-qubit)-cavity system and its analytical solution are introduced. In Sec. 3, the non-locality correlation functions is displayed. We discuss the results of the non-locality correlations in Sec. 4. Finally, we conclude in Sec. 5.

The Physical Model

The studied system is constituted by two sufficiently separated identical artificial two-level atoms (considered as two qubits A and B) resonantly interacting with an open cavity. The two atoms are separated by a distance much larger than their size, consequently their dipole-dipole interplay can be neglected36. In the rotating wave approximation, the total Hamiltonian is:

$$\begin{array}{rcc}\hat{H} & = & {\omega }_{0}{\hat{a}}^{\dagger }\hat{a}+{\omega }_{0}\sum _{i=A,B}{\hat{\sigma }}_{i}^{z}+\sum _{i=A,B}({\lambda }_{i}\hat{a}{\hat{\sigma }}_{i}^{-}+{\lambda }_{i}^{\ast }{\hat{a}}^{\dagger }{\hat{\sigma }}_{i}^{+}),\end{array}$$
(1)

where ω0 represent the qubits and the cavity frequency. \({\hat{a}}^{\dagger }\) and \(\hat{a}\) are respectively the creation and annihilation operators of the cavity mode. The operators \({\hat{\sigma }}_{i}^{\pm }\) and \({\hat{\sigma }}_{i}^{z}\) are the Pauli matrices which are defined by the upper states |1〉i, and lower states |0〉i. λi designate the coupling between the cavity and the qubits. Here, we focus on the case where λi = λ.

If we consider only the dissipative term of the dipole decay of the qubits, the dynamic of the system is given by37

$$\begin{array}{rcl}\frac{\partial \hat{\rho }(t)}{\partial t} & = & -i[\hat{H},\hat{\rho }(t)]\\ & & +\sum _{i=A,B}{\gamma }_{i}([{\hat{\sigma }}_{i}^{-},\hat{\rho }(t){\hat{\sigma }}_{i}^{+}]+[{\hat{\sigma }}_{i}^{-}\hat{\rho }(t),{\hat{\sigma }}_{i}^{+}]),\end{array}$$
(2)

where γi are spontaneous emission rates of the two qubits, which are treated by coupling each qubit to reservoir. In the basis states {|1〉 = |1A1B, n〉, |2〉 = |1A0B, n + 1〉, |3〉 = |0A1B, n+1〉, |4〉 = |0A0B, n + 2〉}, the dressed states, |Ψim〉(i = 1–4), are

$$\,(\begin{array}{c}|{\varPsi }_{1}^{n}\rangle \\ |{\varPsi }_{2}^{n}\rangle \\ |{\varPsi }_{3}^{n}\rangle \\ |{\varPsi }_{4}^{n}\rangle \end{array})=A(\begin{array}{c}\mathrm{|1}\rangle \\ \mathrm{|2}\rangle \\ \mathrm{|3}\rangle \\ \mathrm{|4}\rangle \end{array}),A=(\begin{array}{cccc}{\tilde{a}}_{n} & 0 & 0 & -{\tilde{b}}_{n}\\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0\\ {b}_{n} & -\frac{1}{2} & -\frac{1}{2} & {a}_{n}\\ {b}_{n} & \frac{1}{2} & \frac{1}{2} & {a}_{n}\end{array})\mathrm{}.$$
(3)

where \({a}_{n}=\sqrt{\frac{n+2}{2(2n+3)}},\,{b}_{n}=\sqrt{\frac{n+1}{2(2n+3)}}\) and \(\tilde{a}{(\tilde{b})}_{n}=a{(b)}_{n}\sqrt{2}\). In the case of high −Q cavity (\({\gamma }_{i}\ll {\lambda }_{i}\)), we apply the dressed-states representation (DSR) based on the Hamiltonian eigenstates38,39,40. The operators of the qubits, |1〉ii〈0|, of Eq. (7) are written in the DSR.

In this paper we focus on the case where the two qubits are initially in the excited state (uncorrelated state), i.e., \({\hat{\rho }}_{AB}(0)\) = |1A1B〉〈1A1B|. While the wave function of cavity mode field is initially prepared in the superposition coherent state: |α〉 + κ|−α〉, where α is the intensity of the coherent state, i.e.,

$$\begin{array}{rcl}\,{\hat{\rho }}_{F}(0) & = & \sum _{m,n=0}{Q}_{m,n}|m\rangle \langle n|,\\ & = & \frac{[1+\kappa {(-1)}^{m}][1+\kappa {(-1)}^{n}]}{[1+{\kappa }^{2}+2\kappa {e}^{-2N}]\sqrt{m!n!}}{N}^{\frac{m+n}{2}}{e}^{-N}|m\rangle \langle n|.\end{array}$$
(4)

N = |α|2 designs the mean photon number and κ = 0 and 1 are taken respectively for the coherent state and the superposition coherent state. Coherent states and their superpositions are proposed as major elements for the realization of quantum processing. Using the dressed states space, {|Ψim〉}, of Eq. (3), the initial total density matrix in DSR is rewritten as:

$$\begin{array}{ccc}\,W(0) & = & \sum _{m,n=0}2{f}_{aa}|{\Psi }_{1}^{m}\rangle \langle {\Psi }_{1}^{n}|+{f}_{ba}^{mn}\sqrt{2}[{\hat{h}}_{31}^{mn}+{\hat{h}}_{41}^{mn}]\\ & & +{f}_{bb}^{mn}[{\hat{h}}_{33}^{mn}+{\hat{h}}_{34}^{mn}+{\hat{h}}_{43}^{mn}+{\hat{h}}_{44}^{mn}]+{f}_{ab}^{mn}\sqrt{2}[{\hat{h}}_{13}^{mn}|+{\hat{h}}_{14}^{mn}],\end{array}$$
(5)

where frsmn = Qm,nrmsn(r, s = a, b), \({\hat{h}}_{kl}^{mn}\) = |Ψkn〉〈Ψln|.

In the basis states of the two qubits {|1〉 = |1A1B, n〉, |2〉 = |1A0B, n + 1〉, |3〉 = |0A1B, n + 1〉, |4〉 = |0A0B, n + 2〉}, Eq. 2 becomes

$$\hat{\rho }(t)=\mathop{\sum }\limits_{ij=0}^{4}\mathop{\sum }\limits_{mn=0}^{\infty }{\langle i|{e}^{-i\hat{H}t}W(t){e}^{i\hat{H}t}|j\rangle }_{mn}(t)\,|i\rangle \langle j|\mathrm{}.$$
(6)

To calculate \({\langle i|{e}^{-i\hat{H}t}W(t){e}^{i\hat{H}t}|j\rangle }_{mn}(t)\), we used the following canonical transform; \(W(t)={e}^{i\hat{H}t}\hat{\rho }(t){e}^{-i\hat{H}t}\) in Eq. (7) to became:

$$\dot{W}(t)=\sum _{i=A,B}{\gamma }_{i}{e}^{i\hat{H}t}([{\hat{\sigma }}_{i}^{-},\hat{\rho }(t){\hat{\sigma }}_{i}^{+}]+[{\hat{\sigma }}_{i}^{-}\hat{\rho }(t),{\hat{\sigma }}_{i}^{+}]){e}^{-i\hat{H}t}\mathrm{}.$$
(7)

If i ≠ j, the elements 〈Ψki|W(t)|Ψlj〉 are calculated from

$$\langle {\Psi }_{k}^{i}|W(t)|{\Psi }_{l}^{j}\rangle ={e}^{-({\gamma }_{A}+{\gamma }_{B}){\beta }_{kl}t}\langle {\Psi }_{k}^{i}|W\mathrm{(0)|}{\Psi }_{l}^{j}\rangle ,$$
(8)

with \({\beta }_{12}^{ij}={\beta }_{21}^{ji}=(2{a}_{i}^{2}+\frac{1}{2})\), \({\beta }_{23}^{ij}={\beta }_{32}^{ji}={\beta }_{24}^{ij}={\beta }_{42}^{ji}=({b}_{j}^{2}+\frac{3}{4})\), β22ij = 1, \({\beta }_{13}^{ij}={\beta }_{31}^{ji}={\beta }_{41}^{ij}={\beta }_{14}^{ji}=\) \((2{a}_{i}^{2}+{b}_{j}^{2}+\frac{1}{4})\) and \({\beta }_{33}^{ij}={\beta }_{34}^{ij}={\beta }_{43}^{ij}={\beta }_{44}^{ij}=({b}_{i}^{2}+{b}_{j}^{2}+\frac{1}{2})\).

While, if i = j, the elements 〈Ψki|W(t)|Ψli〉 are calculated from

$$\begin{array}{rcl}{\dot{A}}_{i} & = & 2\gamma {b}_{i}^{2}{B}_{i+1}+\gamma {b}_{i}^{2}{C}_{i+1}-4\gamma {a}_{i}^{2}\,{A}_{i},\\ {\dot{B}}_{i} & = & 2\gamma {a}_{i+1}^{2}{A}_{i+1}+\gamma {b}_{i+1}^{2}{C}_{i+1}-\gamma {B}_{i},\\ {\dot{C}}_{i} & = & 2\gamma {a}_{i+1}^{2}{A}_{i+1}+2\gamma {a}_{i}^{2}{B}_{i+1}+2\gamma {\chi }_{i+1}{C}_{i+1}-\gamma {\varphi }_{i}{C}_{i}.\end{array}$$
(9)

where γ = γA + γB, Ai = 〈Ψ1i|W(t)|Ψ1i〉, Bi = 〈Ψ2i|W(t)|Ψ2i〉, Ci = 〈Ψ3i|W(t)|Ψ3i〉 + 〈Ψ4i|W(t)|Ψ4i〉, \({\chi }_{i+1}=\frac{1}{2}({a}_{i}^{2}+{b}_{i+1}^{2})\) and \({\varphi }_{i}=2{b}_{i}^{2}+\frac{1}{2}\). The Eq. (9) is exactly solvable for the case where each state has at most N photons only, i.e, AN+1 = BN+1 = CN+1 = 0, the case N → ∞ could be considered38. Equation (9) gives at i = N,

$$\begin{array}{rcl}{A}_{i}(t) & = & {e}^{-4\gamma {a}_{i}^{2}t}[{A}_{i}\mathrm{(0)}+\gamma {b}_{i}^{2}{\int }_{0}^{t}[2{B}_{i+1}+{C}_{i+1}]{e}^{4\gamma {a}_{i}^{2}t}dt],\\ {B}_{i}(t) & = & {e}^{-\gamma t}[{B}_{i}\mathrm{(0)}+\,\gamma {\int }_{0}^{t}\,[2{a}_{i+1}^{2}{A}_{i+1}+{b}_{i+1}^{2}{C}_{i+1}]{e}^{\gamma t}dt],\\ {C}_{i}(t) & = & {e}^{-\gamma {\varphi }_{i}t}[{C}_{i}\mathrm{(0)}+2\gamma {\int }_{0}^{t}[{a}_{i+1}^{2}{A}_{i+1}+{a}_{i}^{2}{B}_{i+1}+{\chi }_{i+1}{C}_{i+1}]{e}^{\gamma {\varphi }_{i}t}dt].\end{array}$$
(10)

Therefore \({A}_{N}(t)={e}^{-4\gamma {a}_{i}^{2}t}{A}_{N}(0),\,{B}_{N}(t)={e}^{-\gamma t}{B}_{N}(0),\,{C}_{N}(t)={e}^{-\gamma {\varphi }_{N}t}{C}_{N}(0)\). at i = N − 1, we get

$$\begin{array}{rcl}{A}_{N-1}(t) & = & {e}^{-4\gamma {a}_{N-1}^{2}t}[{A}_{N-1}\mathrm{(0)}\\ & & +\gamma {b}_{N-1}^{2}{\int }_{0}^{t}\mathrm{[2}{B}_{N}+{C}_{N}]{e}^{4\gamma {a}_{N-1}^{2}t}dt],\\ {B}_{N-1}(t) & = & {e}^{-\gamma t}[{B}_{N-1}\mathrm{(0)}+\gamma {\int }_{0}^{t}\mathrm{[2}{a}_{N}^{2}{A}_{N}+{b}_{N}^{2}{C}_{N}]{e}^{\gamma t}dt],\\ {C}_{N-1}(t) & = & {e}^{-\gamma {\varphi }_{N-1}t}[{C}_{N-1}\mathrm{(0)}+2\gamma {\int }_{0}^{t}[{a}_{N}^{2}{A}_{N}+{a}_{i}^{2}{B}_{N}+{\chi }_{N}{C}_{N}]{e}^{\gamma {\varphi }_{N-1}t}dt].\end{array}$$

Then we can calculate for i = N − 2, N − 3, ..., 0. The density operator of the qubits ρAB(t) can be determined, by tracing the cavity field degrees of freedom as:

$${\rho }^{AB}(t)={{\rm{tr}}}_{F}\{\rho (t)\}=\mathop{\sum }\limits_{k\mathrm{=0}}^{\infty }\langle k|\rho (t)|k\rangle \mathrm{}.$$
(11)

Now, we can explore the time evolution of the non-local correlations.

Non-Locality and Concurrence Quantifiers

We adopt as non-locality quantifiers: the trace norm measurement induced non-locality (MIN) and the Bell function. These measures will be compared to the concurrence as a quantifier of entanglement.

Concurrence

The concurrence41 is one of the most used measures of the entanglement between two qubits. It is defined as,

$$C(t)=\,{\rm{\max }}\{0,\sqrt{{\lambda }_{1}}-\mathop{\sum }\limits_{i=2}^{4}\,\sqrt{{\lambda }_{i}}\};\,{\lambda }_{i} > {\lambda }_{i+1},$$
(12)

where λi are the eigenvalues of the following matrix:

$$T={\rho }^{AB}(t)({\sigma }_{y}\otimes {\sigma }_{y}){\rho }^{\ast AB}(t)({\sigma }_{y}\otimes {\sigma }_{y}).$$

Trace-norm MIN

Firstly, the measurement induced non-locality for a two-qubit state ρAB(t) is defined via Hilbert-Schmidt norm (2-norm)27. Unfortunately, just like geometric quantum discord based on 2-norm, it may change under trivial local reversible operations on an unmeasured subsystem of ρAB(t)29. To address this issue, the MIN based on trace norm (1-MIN) and others have been introduced by30,31,32. The modified versions of the p-MIN based on the Schatten p-norm is given by30,31

$${M}_{p}({\rho }^{AB}(t))=\mathop{{\rm{\max }}}\limits_{\chi \in {\varPi }^{A}}{\Vert {\rho }^{AB}(t)-{\Pi }^{A}({\rho }^{AB}(t))\Vert }_{p}^{p}.$$
(13)

where \({\Vert A\Vert }_{p}\) is the Schatten p-norm of a matrix/vector A. Here we use 1-MIN that represents the maximal trace distance between the pre-measurement state and the post-measurement state caused by the locally invariant measurements. the trace-norm MIN in explicit form can be written as:

$$M(t)=\{\begin{array}{cc}\frac{\sqrt{{\chi }_{+}}+\sqrt{{\chi }_{-}}}{2{\Vert \overrightarrow{{\rm{x}}}\Vert }_{1}}, & {\rm{If}}\,\overrightarrow{{\rm{x}}}\ne \mathrm{0;}\\ 2\,{\rm{\max }}\,\{|{r}_{11}|,|{r}_{22}|,|{r}_{33}|\}, & {\rm{If}}\,\overrightarrow{{\rm{x}}}=0.\end{array}$$
(14)

where \({\chi }_{\pm }=\alpha \pm 2\sqrt{\beta }{\Vert \overrightarrow{{\rm{x}}}\Vert }_{1}\), \(\alpha ={\Vert C\Vert }_{2}^{2}{\Vert x\Vert }_{1}^{2}-{\sum }_{i}\,{r}_{ii}^{2}{x}_{i}^{2}\), C = [rii] and \(\beta ={\sum }_{\langle ijk\rangle }{x}_{i}^{2}{r}_{jj}^{2}{r}_{kk}^{2}\), the summation of β runs over all the cyclic permutations of 1,2,3. where where xi = Tr(ρAB(t)(σi ⊗ I)) are the elements of the local Bloch vector x, while, rmn = tr{ρAB(t)(σm ⊗ σn)} represent the components of the correlation matrix R = [rmn]28. \(\overrightarrow{\sigma }=({\sigma }_{1},{\sigma }_{2},{\sigma }_{3})\) represents a vector of the Pauli spin matrices.

If the elements of density matrix ρAB(t) are denoted by: zij = 〈i|ρAB(t)|j〉 = eij + idij, then

$$\overrightarrow{{\rm{x}}}={({e}_{13}+{e}_{24},{d}_{31}+{d}_{42},{z}_{11}+{z}_{22}-1/2)}^{t},$$
(15)

while, the correlation matrix R for a general bipartite quantum state ρAB(t) is given by

$${\bf{R}}=(\begin{array}{ccc}{e}_{23}+{e}_{14} & {d}_{23}-{d}_{14} & {e}_{13}-{e}_{24}\\ {d}_{41}-{d}_{23} & {e}_{23}-{e}_{14} & {d}_{13}+{d}_{24}\\ {e}_{12}-{e}_{34} & {d}_{34}-{d}_{12} & {z}_{11}+{z}_{44}-\frac{1}{2}\end{array})\mathrm{}.$$
(16)

(ii) - Maximum Bell function:

The maximal value of the Bell function Bmax(t), is considered as an indicator of non-locality correlation33. If Bmax(t) > 2, then the Bell’s inequality is violated, i.e., Bmax(t) locates the nonlocal quantum correlations when its value is above 2 (the classical threshold). Here, we use the Bell function defined by

$$B(t)=2\sqrt{{S}_{{\rm{\max }}}}-1,$$
(17)

where, Smax is the summation of the two largest eigenvalues for the matrix \(U={{\rm{R}}}^{\dagger }{\rm{R}}\), R represents the correlation matrix of a two-qubit state ρAB. The function B(t) identifies NLC when it is above the classical threshold 1.

NLC Dynamics

Effect of the coherent cavity superposition

In Fig. 1, we display the resulted non-locality correlations of the two qubits with respect to the unitary interaction γ = 0. Where the trace-norm MIN, M(t), maximal Bell function, B(t), and the concurrence C(t) are displayed for different values of the superposition parameter κ = 0; κ = 0 in (a), κ = 1 in (b) and κ = −1 in (c) with the initial coherence intensity N = 2.

Figure 1
figure 1

Time evolutions of M(t) (dashed dotted plots), B(t) (dashed plots) and C(t) (solid plots) with the dissipative rate γ/λ = 0 and N = 2 for different values of κ = 0 in (a), κ = 1 in (b) and κ = −1 in (c).

Full size image

From a chosen initial pure state of the qubits |1A1B〉 or |0A0B〉, the elements of the correlation matrix R are zero except for r33 = 1, and the Bloch vector is x = (0, 0, ±1)t ≠ 0. Therefore, M(0) = 0, B(0) = 1, and C(0) = 0, i.e, the state of the qubits does not have correlations. If the pure state of the qubits develops to one of the maximal correlated states, \(\frac{1}{2}\mathrm{(|01}\rangle \pm \mathrm{|10}\rangle )\), the elements the correlation matrix R are zero except for the element r11 = ±0.5, r22 = ±0.5 and r33 = −0.5, and the Bloch vector is x = (0, 0, 0)t. Therefore, M(0) = 1, B(0) = 1.8284, and C(0) = 1 (generating maximal NLCs). Otherwise, the qubits-cavity interaction generates partial correlations.

In Fig. 1a, we consider the initial coherent state κ = 0 while the dipole decay of the qubits is neglected (γ = 0). We observe that the unitary interaction leads to: (1) the uncorrelated state of the qubits, |1A1B〉, produces an oscillatory partial entanglement and non-locality correlations during the time evolution of the qubits-cavity interaction. These partial correlations are enhanced with increasing the interaction time. As the unitary interaction evolves, the concurrence C(t) is zero for a short time, and it suddenly grows to its partial maximum value at particular points. These points are called growth-start points (GSPs). C(t) presents sudden birth and death entanglement42.

(2) the trace-norm MIN M(t) grows from zero (i.e., GSP is zero) to its partial maximum values. It has different behaviour compared to C(t). M(t) never vanishes. The upper bounds of trace-norm MIN are larger than that of the concurrence. (3) with respect to B(t), we observe that the Bell’s inequality is violated for short time intervals, in which B(t) > 1.

Where, in the case of κ = −1, the values of NLC functions may reach approximately the values of the maximal correlated states, \(\frac{1}{2}\mathrm{(|01}\rangle \pm \mathrm{|10}\rangle )\), that are mentioned above. This mean that there is relation between the correlation dynamics and the evolution of the state of the qubits.

In Fig. 1b,c, the dependence of the NLCs on the superposition of coherent states parameter κ is depicted. The NLC functions for the two cases of the initial even coherent state κ = 1 in (b) and the initial odd coherent state κ = −1 in (c) are simulated with the same data of Fig. 1a.

We observe that the generated NLCs are bigger compared to the ones of the coherent cavity. The superposition of the coherent state parameter κ leads to the increase of the time intervals of the maximal violation of Bell’s inequality of B(t) > 1.

Effect of the dipole qubits decay

In Fig. 2, the NLC functions M(t), B(t), and C(t) are plotted for the initial coherent state κ = 0 and the dipole decay of qubits (γ = 0.08λ). We note a more rapid deterioration of the non-local correlations. In case of coherent cavity field κ = 0 with γ/λ = 0.08, we observe: (1) due to the dipole decay of the qubits, the two functions of maximal Bell function and the concurrence vanish approximately whereas the trace norm MIN function does not vanish.

Figure 2
figure 2

As Fig. 1, but with γ = 0.08λ.

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For the cases of the initial even coherent κ = 1 and odd coherent κ = −1 microcavity field, we observe that the oscillations, amplitudes and the negativity of the NLC functions M(t), B(t), and C(t) are more robust against the rate of dipole qubits decay γ/λ, (see Fig. 2b,c).

Therefore, the generated NLCs depend on the dipole decay and on the initial coherent cavity field. Due to the dipole qubits decay, the stable state of the two qubits has a value of the trace norm MIN correlation beyond that of entanglement and non-locality Bell-correlation.

Effect of the initial coherence intensity

In Fig. 3, we analyze the effect of the initial coherence intensity N, where N = 0.5 is small and the dipole decay of qubits is neglected. We notice the change of the dynamical behavior of the NLCs. We observe that the NLC functions M(t), B(t), and C(t) (see Figs. 1 and 3) exhibit extreme values and amplitudes less than that of N = 2.

Figure 3
figure 3

As Fig. 1(a), but with α = 0.5.

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The Bell’s inequality is violated during short intervals, for the initial even coherent κ = 1 and odd coherent κ = −1 cavity field. However, for a coherent state there is no violation of the Bell’s inequality is observed. We deduce that the amount of the generated entanglement and non-locality correlations may be increased by increasing the initial coherence intensity N.

Figure 4 shows the effect of the dipole decay γ = 0.08λ on NLCs between the two qubits. We observe that the NLCs for γ = 0.05λ have damped oscillations, their amplitudes decrease until completely vanish. When the dipole decay is increased, reduction of the final disappearance time of the NCLs is noted. For the small initial coherence intensity, N = 0.5 (see Fig. 4b), C(t) is quasi-periodic. The sudden birth and death entanglement is observed for large time windows. The generated NLCs are weak and has less robustness as the initial coherence intensity decreases. The NLCs are reduced by the decrease of the initial coherence. The extreme values of trace norm MIN is more robust than the entanglement and non-locality Bell-correlation.

Figure 4
figure 4

As Fig. 1, but for α = 0.5 with γ = 0.08λ.

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Conclusion

Throughout this paper, an analytical description of a cavity contains two qubits spatially separated is established. The non-locality correlations [including trace norm measurement induced non-locality, maximal Bell-correlation and concurrence entanglement] of the two qubits are explored via the trace norm measurement induced non-locality and the Maximum Bell function. The rise in two-qubit damping rates induces a fast deterioration of the coherence. We notice that this system presents sudden birth and death entanglement. The generated non-locality correlations essentially determined by the loss rate of the two-qubit and the initial coherence of the cavity.