Atomic non-locality dynamics of two moving atoms in a hybrid nonlinear system: concurrence, uncertainty-induced non-locality and Bell inequality

The paper explores the non-locality dynamics of two moving atoms in a hybrid nonlinear system, which describes a nondegenerate parametric amplifier cavity containing two moving atoms with Stark shift interactions. Based on the Bell function, uncertainty-induced non-locality and concurrence, the two moving atomic non-locality is calculated numerically, when the cavity is filled initially by pair coherent fields. The results showed that the hybrid nonlinear qubit-cavity interactions have a high ability to generating quantum correlations. The amount of the generated quantum correlation can be improved and accelerated significantly with the increase of the time-dependent qubit location parameter. The increase of the Stark shift ratio and the difference between the mean-number photons of the two cavity fields leads to a clear deterioration of the generated quantum correlation.


Introduction
The generation of quantum information resources (as: coherence, entanglement and other types of quantum correlation) due to hybrid nonlinear interactions between two-level atomic systems (qubits) are highly substantial for quantum technologies (Modi et al. 2012). Quantum entanglement (QE) is the first type of the quantum correlations between different constituents of multipartite quantum system (Nielsen and Chuang 2000). In quantum information (QE) is an important physical resource and have various potential applications is several quantum approaches as computing (Nielsen and Chuang 2000), communications (Bouwmeester et al. 1997) and key distribution (Ekert 1991). Recently, It is emphasized that all quantum correlated states need not be entangled states (Lanyon et al. 2008;Datta and Vidal 2007). Therefore, there are some quantum information protocols can be implemented with separable mixed states (Horodecki et al. 2009). Consequently, some other quantifiers were introduced to detect QCs beyond entanglement as: measurement-induced disturbance (Luo 2008), quantum discord (Ollivier and Zurek 2001) and geometrical measures which based on p-norm (Dakic et al. 2010) and skew information quantity (Girolami et al. 2013). QCs can be applied in various branches of quantum engineering, quantum cryptography and quantum information (Nielsen and Chuang 2000;Horodecki et al. 2009;Mohamed 2017). This work aims to investigating the quantum correlation of uncertainty-induced non-locality (UIN) (Wu et al. 2014) beyond concurrence entanglement and Bell nonlocality. UIN is primarily defined by the skew information quantity. The UIN has been introduced as another nonclassical quantifier (Mohamed and Eleuch 2018;Mohamed et al. 2020;Abdel et al. 2020;Zhao et al. 2020;Abd-Rabbou et al. 2019;Mohamed et al. 2021;Abo-Kahla et al. 2021) and it considered as a revised version of measurement-induced non-locality with contractively.
Research on the quantum information resources present in hybrid nonlinear systems allows the adding of a new physical parameters which contribute to the generation of the quantum correlations beyond entanglement, which are very useful for quantum information (Mohamed et al. 2019a, b;Abdalla et al. 2017a, b). The hybrid nonlinear systems can be realized based on the superconducting circuits, ion traps and quantum dots (Baghshahia et al. 2014). The hybrid nonlinear qubit-cavity interactions resulted by considering the mode structure of the cavity field (which is leads to atomic motion and Stark shift terms) (Sukumar and Buck 1981;Li and Peng 1995) are very important to performance and efficiency in quantum information (Li and Peng 1995). In practical experiences, the two-level atomic systems cannot always be regarded to be static during their interaction with the cavity fields. Therefore, the time-dependent two-level atomic location are logically considered. The effect of the time-dependent twolevel atomic location has been realized through an atom interacting with a cavity (Rempe et al. 1987). It is found that the time-dependent qubit-cavity interactions can be enhanced the nonclassicality of the system and the dynamics of entanglement (Ghorbani et al. 2017;Xiao-Juan and Mao-Fa 2008;. Another important and widely used concept is the entanglement pair coherent state (PCS), which is very useful for primary tasks in quantum information theory (Zukowski et al. 1993;Bennett et al. 2001;Duan et al. 2001), such as teleportation, Bell nonlocality, and non-classical features of sub-Poissonian distribution and squeezing (Agarwal and Biswas 2005;Obada and Khalil 2010). PCSs can be realized by different nonlinear processes in a two-photon medium (Agarwal and Biswas 2005).
The manuscript is organized as follows: The model of the hybrid nonlinear system and its description are presented in the second section. In third and fourth sections, the definition and the dynamics of the atomic non-locality quantifiers (the Bell function, uncertaintyinduced non-locality and concurrence) are illustrated. Finally, in fifth section, the our results are concluded.

The considered model
The considered model a hybrid nonlinear system describes a nondegenerate parametric amplifier cavity containing two effective moving atoms (each atom represents qubit with two upper and lower states) with Stark shift interactions (Alqannas and Khalil 2018). The qubit-field Hamiltonian can be written as with the two-field operators ̂ † i and ̂i(i = 1, 2) and the two-qubit location function described by f (t) = cos lt , where l represents the number of half wavelengths of the cavity field. N k =̂ † k̂k represent the mean-number photons of the two cavity fields. ̂k ± and ̂( k) z represent two-qubit Pauli operators, is the frequency of the qubit-field resonance case. represents the qubit-field interaction coupling. The Stark shift intensity S i represent the shifting of the energy levels of atoms due to the applied cavity fields (Obada et al. 2013, xxx).
To find a particular solution for the schrodinger equation that describes the dynamics of the atom-cavity state based on Eq.1, we assume that the initial qubit-field state as, where �1 A , 1 B ⟩ denotes to the upper two-qubit states. The two-mode fields are initially in the pair coherent state (Agarwal and Biswas 2005) with the distribution function, where is the eigenvalue of the operator ̂1̂2 , and | | 2 represents the initial coherent intensity of the two-mode cavity fields.
Δ n is the difference between the mean-number photons of the two cavity fields. In the space of the atom-field system, where , the general solution for the qubit-field state � (t)⟩ takes the form By using the schrodinger equation, the dynamics of the atom-cavity state are obtained by the following differential equations, The differential equations of Eq.4 will be calculated numerically. The numerical twoqubit reduced density matrix of the final time-dependent state � (t)⟩ are used to analyze the two-qubit correlations based, which is given by In the following, the two-qubit correlations will be investigated via the atomic concurrence entanglement, atomic uncertainty-induced non-locality and atomic Bell function non-locality.

Two-qubit correlation quantifiers
• Two-qubit concurrence: The generated two-qubit entanglement due to the timedependent qubit-cavity interactions can be investigated by concurrence (Bennett et al. 1996). The concurrence is one of the most common measure of the entanglement between two particles A and B with the density matrix ̂A B (t) . The concurrence is defined as • Uncertainty-induced non-locality : For a bipartite quantum state AB and a local observable K, the quantity of the skew information is given by Girolami et al. (2013), By using this quantity of the skew information, the uncertainty-induced non-locality (UIN) was introduced as another correlation quantifier to investigate the generated twoqubit correlation. For the two-qubit reduced density matrix ̂A B (t) , the UIN is given by Wu et al. (2014) that means that, U( AB (t)) quantifier represents the maximal skew information between the local commuting observable K and the two-qubit reduced density matrix. Based on the maximum eigenvalue min of the symmetry 3x3-matrix W AB = [w ij ] , the explicit form of UIN is given by Wu et al. (2014): where � ⃗ is the Bloch vector and where i , i = 1, 2, 3 are the Pauli matrices. • Two-qubit Bell non-locality : Here, we use the maximal values of the two-qubit Bell function B max (t) as an indicator of two-qubit non-locality (Brunner et al. 2014;Horodecki et al. 1995). This two-qubit non-locality satisfies: B max (t) > 2 , i.e., the Bell-Chsh inequality is violated. The analytical expression of the two-qubit Bell function B max (t) is given by where i (i = 1, 2) are the two largest eigenvalues of the matrix T † T , T = [t ij ] is the correlation matrix (Horodecki et al. 1995), its elements are defined by t ij = tr{ AB (t) (1) i (2) j } , i, j = 1, 2, 3 . In our investigation, the Bell non-locality is quantified by, The moving atomic state has Bell non-locality when this function satisfies B(t) > 1.

Atomic non-locality dynamics
We study the effect of time dependence and Stark shift on the measures C(t), B(t) and U(t). Fig.1, displays the generation of the atomic moving non-locality via the Bell function, uncertainty-induced non-locality and concurrence in the presence of the nonlinearity of the Stark shift interaction with different atomic locations. First, we exclude the dependence on time ( l = 0 ). For small effective of the Stark shift S = 1 , the quantities C(t), B(t) and U(t) oscillate with the same behavior. The maximum and the minimum values of entanglement that predicted by both quantifiers are displayed as the same time. The three measures fluctuate chaotically between the maximum and minimum values. The phenomena of sudden death and sudden birth are realized almost every 2 , as seen in Fig.1a. The random oscillations become regular after adding the time dependence ( l = 1 ). Therefore, the entanglement between the two qubits improves frequently and regularly. The quantities C(t), B(t) and U(t) start from pure state, then partial entanglement is generated until the functions C(t), B(t) and U(t) reach their maximum values. The phenomena of sudden death and sudden birth are realized every .
From Fig.1b and c, we can find that the generated quantum correlations between the two qubits improves with increasing l (the time dependence parameter). The intensity of the oscillations increases, so the phenomena of sudden birth and sudden death are realized every 2 . Moreover, the entanglement periods between the parts of the quantum system improved with increasing the parameter l. In general, the upper bounds of the amount of entanglement that shown by the concurrence are much larger than those quantified by the uncertainty-induced non-locality. (10) In Fig.2, the effect of the Stark shift ratio S on the dynamics of atomic moving nonlocality is shown. Consider S = 1.5 , to activate the Stark shift role. Difference of the Stark shift from one reduces the entanglement between the two qubits. We deduce from the results that both C(t) and B(t) are affected by the Stark shift terms, so the maximum values decreased significantly. While the upper bounds of U(t) is not affected by the presence of Stark shift, as evident in Fig.2a.
From Fig.2b we can find that the C(t) and B(t) measures are strongly influenced by increased Stark shift effect. The entanglement between the two qubits is greatly reduced. While the maximum and minimum values of the measure U(t) are almost unaffected by the presence of the Stark shift as seen in Fig.2b. The relationship 10 indicates that the measure U(t) depends on the minimum values of AB , so the Stark shift change is related to the maximum values of AB . Therefore, the effect of Stark shift on measure U(t) is not demonstrated. Fig.3 show the effect of the difference Δ n between the mean photon number of the two cavity fields on the dynamics of atomic moving non-locality. The amount of entanglement decreases significantly after adding the parameter Δ n . The phenomena of sudden death and sudden birth improve greatly. However, the oscillation amplitudes decrease and consequently, the maximum bounds of the entanglement that predicted via both quantifies decreases. The concurrence measure collapses by increasing the parameter Δ n . The measures U(t) and B(t) are small affected by the increase of the parameter Δ n . We can deduce that the increase of the parameter Δ n resists the generated atomic correlations. Fig.4 illustrates the effect of the intensity coherent cavity fields of their mean photon numbers, with the large value for the -parameter, on the dynamics of atomic moving non-locality. Increasing the mean photon numbers (initial coherent intensity of the two-mode cavity fields) leads to increase the generated quantum correlation between the two qubits. We note that the amplitudes and stability of the atomic concurrence entanglement, atomic uncertainty-induced non-locality as well as the atomic Bell non-locality can be enhanced with the increase of the initial coherent intensity of the two-mode cavity fields.
From Fig.4b, we can find that the quantum correlation measures are affected after considering the Stark shift an increase in the parameter . The Stark shift effect is more pronounced. Whereas, a weak entanglement is generated. The phenomena of sudden death and sudden birth disappear after considering the Stark shift with the exception of some points. Therefore, one can say that the initial coherent intensity of the two-mode cavity fields can be used to a generator factor of the generated quantum correlation between the atomic system.
The results of the quantifiers C(t), B(t) and U(t) show that the hierarchy (Costa et al. 2016;Gomes and Angelo 2018) between the Bell-nonlocality and the concurrence-entanglement is satisfied, meaning that Bell-nonlocality implies entanglement, which in turn implies the uncertainty-induced non-locality. The rigid hierarchy between the Bell-nonlocality and the quantum correlation via entanglement reported (Costa et al. 2016;Gomes and Angelo 2018), explicitly. But, the hierarchy between the concurrence and the uncertainty-induced non-locality (that is entanglement measure for pure state states Girolami et al. 2013;Wu et al. 2014) is not reported explicitly. It because that the entanglement measures have ordering difficulties (Wei 2003).

Conclusions
In this paper, we have considered a hybrid nonlinear system describes a nondegenerate parametric amplifier cavity containing two effective moving atoms with Stark shift  interactions. The general solution is obtained numerically, to study the effect of the time dependence and the Stark shift parameters on the generated atomic correlations of the Bell function, the uncertainty-induced non-locality and the concurrence. The results showed that the amount and the stability if the generated atomic correlations have high sensitivity to the parameters of the Stark shift, the qubit-cavity time-dependent coupling, the initial coherent intensity of the two-mode cavity fields. The concurrence entanglement is severely affected when activating the role of Stark shift parameter. The Bell-nonlocality, the UIN-correlation, and the entanglement improved significantly with the increase of the qubit-cavity time-dependent coupling, which plays an important role in generating quantum correlation between the two systems. Increasing the initial coherent intensity of the two-mode cavity fields generates more entanglement. Whereas, the increase of the parameter of the difference between the mean-number photons of the two cavity fields leads to a clear deterioration of the generated atomic correlations.