Non-locality Correlation in Two Driven Qubits Inside an Open Coherent Cavity: Trace Norm Distance and Maximum Bell Function

We analytically investigate two separated qubits inside an open cavity field. The cavity is initially prepared in a superposition coherent state. The non-locality correlations [including trace norm measurement induced non-locality, maximal Bell-correlation, and concurrence entanglement] of the two qubits are explored. It is shown that, the generated non-locality correlations crucially depend on the decay and the initial coherence intensity of the cavity field. The enhancement of the initial coherence intensity and its superposition leads to increasing the generated non-locality correlations. The phenomena of sudden birth and death entanglement are found.

against decoherence 34 . 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 discord 30,31,35 .
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 neglected 36 . In the rotating wave approximation, the total Hamiltonian is: where ω 0 represent the qubits and the cavity frequency. ˆ † a and â are respectively the creation and annihilation operators of the cavity mode. The operators σ ± i and σ i z are the Pauli matrices which are defined by the upper states |1〉 i , and lower states |0〉 i . λ i designate the coupling between the cavity and the qubits. Here, we focus on the case where λ i = λ.
If we consider only the dissipative term of the dipole decay of the qubits, the dynamic of the system is given by 37ˆˆˆˆˆˆˆˆ∑ ρ ρ γ σ ρ σ σ ρ σ where γ i are spontaneous emission rates of the two qubits, which are treated by coupling each qubit to reservoir. In the basis states In the case of high −Q cavity (γ λ  i i ), we apply the dressed-states representation (DSR) based on the Hamiltonian eigenstates [38][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., ρˆ(0) AB = |1 A 1 B 〉〈1 A 1 B |. 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.,   www.nature.com/scientificreports www.nature.com/scientificreports/ In the basis states of the two qubits . The Eq. (9) is exactly solvable for the case where each state has at most N photons only, i.e, A N+1 = B N+1 = C N+1 = 0, the case N → ∞ could be considered 38 . Equation (9) gives at i = N, Then we can calculate for i = N − 2, N − 3, ..., 0. The density operator of the qubits ρ AB (t) can be determined, by tracing the cavity field degrees of freedom as: 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 concurrence 41 is one of the most used measures of the entanglement between two qubits.
It is defined as, where A 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:  28 . σ σ σ σ → = ( , , ) 1 2 3 represents a vector of the Pauli spin matrices. If the elements of density matrix ρ AB (t) are denoted by: z ij = 〈i|ρ AB (t)|j〉 = e ij + id ij , then while, the correlation matrix R for a general bipartite quantum state ρ AB (t) is given by

(ii) -Maximum Bell function:
The maximal value of the Bell function B max (t), is considered as an indicator of non-locality correlation 33 . If B max (t) > 2, then the Bell's inequality is violated, i.e., B max (t) locates the nonlocal quantum correlations when its value is above 2 (the classical threshold). Here, we use the Bell function defined by max where, S max is the summation of the two largest eigenvalues for the matrix = † U R R, R represents the correlation matrix of a two-qubit state ρ AB . The function B(t) identifies NLC when it is above the classical threshold 1.
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, |1 A 1 B 〉, 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 entanglement 42 .
(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. , 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 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. www.nature.com/scientificreports www.nature.com/scientificreports/ 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.  Fig. 1(a), but with α = 0.5.

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.