Influence of deformed cavity field and atomic dipole interaction on the quantum correlations of two-qubit system

The possibility of generating of some quantum correlations of the entangled or separable initial two-qubit system interacting with a deformed cavity mode in the presence of dipole-dipole interaction is discussed. These quantum correlations are the coherence, non-separability and steering degrees which are quantifiers by using l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ l_1 $$\end{document} norm of coherence, Entanglement of Formation and Einstein–Podolsky–Rosen -steering. It is shown that maximizing or minimizing the quantum correlations depends on the initial setting states of the two-qubit system. To maximize the quantum correlations on the presence of deformation, one has to prepare the initial two-qubit system in an entangled state, setting large values of the coupling dipole and decreasing the effect of the deformation parameter.


Introduction
Quantum information generation induced by the interaction between a quantized field and a low-dimensional atomic system is one of the hottest topics in quantum theory (Ripoll 2022). Rabi model is the first model that described the interaction between the atomic system and a quantized field where it investigated the coherence between a two-level atom and an ideal cavity field (Rabi 1937). Jaynes-Cummings (JC)-model is another simple model that described the atom-field interaction under the rotating wave approximation (Jaynes and Cummings 1963). From then on generalizations began of the JC model including the generalizations of either the quantized field or the atomic system or all of them. For example, the information generation induced the interaction between multi-photon JC-model and two-level atom is discussed (Abdel-Hafez and Obada 1991). The entanglement and non-classical correlation of a moving two-level atom and multi-photon of cavity field in the presence of classical field and Kerr-like medium are studied (Obada et al. 2012a;Abd-Rabbou et al. 2019). The coherence and tomographic entropy of the non-linear SU(1,1) and SU(2) quantum system are investigated (Abd-Rabbou et al. 2022a). Recently, the effect of external environment on the atom-field interaction such as, the Stark shift (Obada et al. 47 Page 2 of 13 2012b, 2013; Ghasemian and Tavassoly 2021), vibration graphene sheet (Alotaibi et al. 2021) and optomechanical cavity (Hao et al. 2019;Alotiabi et al. 2022) are examined. Moreover, the interaction between a cavity field and high-dimensional atomic system is demonstrated, such as three level atom (Eftekhari et al. 2022), four-level atom , five-level (Devi et al. 2022) and N-level atom (Ibrahim et al. 2022).
On the other hand, the effect of a deformed field is paid more attention to by theoretical physicists. The deformed Heisenberg algebra with set of control parameters are used in some physical applications (Lavagno 2008;Haghshenasfard and Cottam 2012;Bukweli-Kyemba and Hounkonnou 2012;Fakhri and Sayyah-Fard 2021). Moreover, All algebra of quantum harmonic oscillators of Boson creation and annihilation operators can be extended by f-deformed algebra (Man'ko et al. 1997). The dynamics of quantum entanglement and some non-classical properties of a deformed system consists of atomic-field interaction with Kerr-like medium are investigated (Chatterjee 2019). The influence of the parity-deformed cavity photons on the entanglement transfer to distant atomic system are discussed (Dehghani et al. 2019). The enhancement of purity cost, regularized fidelity and accuracy of information transfer for the two-qubit system inside a deformed cavity field are analyzed (Metwally et al. 2010). For a large value of deformation, the entanglement of twoqubit system that interacting with a cavity deformed mode with multiphotons are studied (Metwally 2011).
As far as we know, the quantum correlations of the quantum systems have a significant role in quantum information theory and its implementations (Nielsen and Chuang 2000). Among these quantum correlations is quantum coherence, which quantifies by using a family of coherence measures. Linear entropy (Zurek et al. 1993), l 1 norm of coherence (Zhao et al. 2019), relative entropy (Baumgratz et al. 2014), and Jensen-Shannon divergence (Radhakrishnan et al. 2016) are used to quantify the coherence degree. The nonseparability of quantum states (entanglement) is another quantum correlation, which is all applications depend on the robustness of entanglement between subsystems (Plenio and Virmani 2014). Entanglement of formation (Wootters Mar 1998), concurrence (Coffman et al. 2000) and negativity (Verstraete et al. 2001) are employed to measure the non-separability of quantum states. Recently, quantifying the possibility of steering between quantum subsystems represents one of the most important quantum correlations (Skrzypczyk et al. 2014). It is applied as a measure of quantum correlation for some quantum systems like, two-qubit system (Costa and Angelo 2016;Xiao et al. Apr 2017), qubit-qutrit system (Abd-Rabbou et al. 2022) Heisenberg chain model (Cheng et al. 2021; and atomic-field interaction (Wang et al. 2022). Some nonclassical correlations dynamics of dipole-dipole coupled two qubits within the differences between trace distance discord and entanglement taking into account the initial extended Werner-like states were been investigated (Zheng-Da et al. 2015;Khedif and Daoud 2018). The non-classicality and the non-locality of two-qubit Werner state and two-qutrit under the effect of some classical noise environment have been discussed (Rahman et al. 2022a, b). Some quantum correlations dynamics features of Bell nonlocality, entanglement and entropic uncertainty of Dzyaloshinskii-Moriya (DM), Kaplan-Shekhtman-Entin-Wohlman-Aharony(KSEA) interactions with external magnetic field, and intrinsic decoherence have obtained (Abd-Rabboul et al. 2021;Hashem et al. 2022;Essakhi et al. 2022). The concurrence is a simple measure of entanglement that provides an analytic formula for the entanglement of formation (EoF) which is one of three widely studied measures of entanglement of a general bipartite system (Wootters 2001).
In this paper, we are motivated to investigate some quantum correlations of a physical system consists of a two-qubit system interacts locally with a deformed cavity photon in the presence of dipole-dipole interaction. These quantum correlations are studied by employing the l 1 norm of coherence. Entanglement of Formation (EoF), and Einstein-Podolsky-Rosen (EPR)-steering. In this context l 1 norm of coherence are used as a quantifier of coherence degree, where it plays a pivotal role in quantum experimental and biological systems (Lloyd 2011;Kang-Da et al. 2021). EoF are employed to measure the non-separability of our quantum system, where it is important for implementing the quantum computer and its application (Noel et al. 2022), Finally, we are use the quantum steering for deep study of quantum correlation, where the steering is subset of quantum entanglement (Zhao et al. 2020). Therefore, the paper is organized as: Sect. 2 is devoted to propose our physical model and demonstrate the exact solution system. The mathematical quantum quantifiers are presented in Sect. 3. Finally, our results are summarized in Sect. 4.

The suggest physical model
Let us assume that a physical Hamiltonian model including two-qubit system (a pair of two-level atoms) interacts locally with a quantized deformed cavity field in the presence of dipole-dipole interaction. In the rotating-wave approximations, the physical model can be written as, where f ( Ω i ) denotes the frequency of deformed cavity (two-qubit) system, is the coupling strength between the deformed field and two-qubit system, and g is the coupling intensity of dipole-dipole interaction. ̂( i) z , and ̂( i) ± are the Pauli spin and ladder operators, while Â † and Â are the generalized deformed creation and annihilation Boson operators, linked to well known bosonic operators â † and â in which, These operators obey, where f(n) is an arbitrary deformed function. In this paper, we are considered the function f(n) subject to the q-deformation (Arik and Coon 1976), which is given by, where q is the controller parameter for the deformation rate.
The main task of this manuscript is to discuss the influence of the dipole-dipole interaction and deformed field on the temporal evolution of atomic quantum coherence, non-separability and steering. Therefore, it is important to obtain the states of two atomic subsystems, where it is obtained as ̂A B = Tr field � (t)⟩⟨ (t)� . In this case, the reduced density matrix of the atomic subsystem is given by,

Mathematical forms and numerical results
In this section, we introduce the mathematical forms of l 1 norm of coherence, the entanglement of formation, and steering. The effects of ideal and deformed cavity field and dipoledipole interaction on the coherence, non-separability and steering are discussed.

l 1 norm of coherence
In this subsection, we will use l 1 norm of coherence to illustrate the coherence between two qubits along the interaction time. It is considered one of the most easier quantum quantifiers, which it obtains by using the absolute value of the off diagonal elements of the atomic density matrix (14). Mathematically, the l 1 norm of coherence is defined as Zhao et al. (2019), where C l 1 = 0 for a maximum decoherence degree. In Fig. 1, we employ the final atomic density operator 14 to display the temporal evolution of coherence, where we consider two different initial setting states of the atomic system. The first structure is assumed that the two atoms are prepared in a product state, where each atom is in an excited states. This structure is obtained by setting a 1 = 1 and a 2 = 0 = a 3 = a 4 . Explicitly, this state can be written as � (0)⟩ = �11⟩ . The second suggested structure of the atomic system is that the two atoms are initially prepared in the maximum entangled state, where we set a 1 = a 4 = 1 √ 2 and a 2 = a 3 = 0 , consequently � (0)⟩ = 1 2 (�11⟩ + �00⟩) . Overall, the strength of atomic frequency Ω i and deformed field f are equal 1, while the intensity of initial cavity field = 5.
In this figure we investigate the behavior of C l 1 on the absences of the deformation, i.e., we set f (n) = 1 . It is clear that the behavior of C l 1 depends on the initial structure of the atomic system (product/maximum entangled). If the atomic system is initially prepared in a maximum entangled state, the upper bounds of the temporal coherence degree are larger than those displayed when the atomic system is prepared in a product state. However, as scaled time increases, the coherence degree for the initial maximum entangled state is less than those depicted at the initial product state. Moreover, at the onset interaction t = 0 , the function C l 1 = 0 for the initial atomic product state, while C l 1 = 1 for the initial maximum entangled state. The upper bounds of C l 1 increase and shifted as the coupling strength of the dipole increases.
The influence of the deformation function is displayed in Fig. 2 and different values of the deformation parameter q are considered. The general behavior shows that as the deformation parameter q increases, the coherence degree decreases with the interaction time increase. For the initial atomic product state (Figs. 2a, b), it is clear the number of oscillations of the function C l 1 increase as the deformation parameter q increases, where 0 ≤ C l 1 ≤ 3 which generally given by Fig. 2 The temporal evolution of C l 1 against the scaled time, where f (n) = √ 1−q n n(1−q) , g = 1 (solid curve) and g = 10 (dash curve), a, b the initial atomic state in excited state with q=0.1 and 0.8 respectively. c, d the initial atomic state in maximum entangled state with q=0.1 and 0.8 respectively quantum state (Zhao et al. 2019). The oscillations at a small value of coupling dipole are regular than that is disclosed for large coupling. Moreover, the maximum and minimum bounds increase as the deformation rate increases. Different behaviour can be seen in Fig. 2c and d, where the atomic system is initially prepared in the maximum entangled state. At a small deformation rate, the function C l 1 chaotically osculates between the minimum bound 1 and upper bound 3 for a small coupling dipole, while the maximum bound decreases at a large value of coupling dipole. On the other hand, as the deformation rate and interaction time increase, the maximum bounds of C l 1 at large values of coupling dipole intensity are larger than those displayed at small values.

Entanglement of formation
In this subsection, we shall discuss the non-separability between qubits A and B. The EoF is one of the significant indicators of non-separable states, which is quantified based on the number of resources required to generate a minimum average of entanglement degrees (Wootters Mar 1998). For a two-dimensional bipartite quantum system, the EoF can be explicitly expressed as, , and the function C( ) is the concurrence of a two-qubit state, which is defined by Wootters (Mar 1998), here i are non-negative eigenvalues of the matrix ( y ⊗ y ) * ( y ⊗ y ) , which are sorted in the decreasing order.
In Fig. 3, we are applying Eq. (16) to evaluate the non-separability of the atomic subsystems against the interaction time, where an ideal cavity is considered and either an initial maximum entangled state or pure state is assumed. It is clear that preparing the atomic system in a product state, the possibility of generating a non-separable state at a small g is smaller than that displayed at large g. However, preparing the atomic system in a maximum entangled state and large coupling of dipole strength leads to generating a separable state with a long interaction time. From this figure, one can observe that the non-separability is predicted at a long interaction time with the large intensity of coupling dipole and the initial atomic product state.

(b)
Scaled Time E Fig. 3 The temporal evolution of E against the scaled time with the same parameters as Fig. 1 47 Page 8 of 13 Moreover, the initial maximum entangled state turns into a partially entangled state or separable state according to the intensity of the coupling dipole at the point of interaction time.
The effect of the deformed cavity on the degree of non-separability by using the EoF is displayed in Fig. 4, where large and small values of deformation rate (q) and coupling dipole g are taken into our account. At the initial product state, Fig. 4a and b show that the function E osculates between a maximum entangled state and a separable state. The intensity of g has a slight effect compared with that displayed in Fig. 2, while the intensity of q generates a chaotic behaviour and reduces the maximum bounds of a non-separability. For the initial maximum entangled state, Fig. 4c and d disclosed that the atomic system regularly osculates between a maximum entangled state and non-separability state at a small deformed rate. The minimum bounds of E with a large g are lower than those depicted with a small g. By increasing the deformation rate, the initial maximum entangled state switches into a partially entangled state. From this figure, one can remark that the large intensities of coupling dipole and deformation rate dissipate the entanglement degrees at the initial maximum entangled state. Moreover, it has a slight effect on the initial product state.

Steering
In this subsection, we employ the final density matrix (14) to quantify the steerability degree from qubit A to qubit B by using the EPR-steering. The inequality of EPR-steering is reconstructed based on the conditional entropic Heisenberg uncertainty principle for the discrete observables as Skrzypczyk et al. (2014), Costa and Angelo (2016), (c) (d) Fig. 4 The temporal evolution of E against the scaled time with the same parameters as Fig. 2 where H(•) = − ∑ i p i log 2 p i is the Shannon entropy with the probabilities p i of the system. H(B|A) = H( AB ) − H( A ) where the discrete observables A and B. For more details see Ref. Abd-Rabbou et al. (2022b). By normalized I AB , one can obtain the degree of steerability as Costa and Angelo (2016), Cheng et al. (2021), where I max AB is steering of maximum entangled Bell state, which equal 6. For a non-steerable state, S = 0 , while S = 1 for a maximum steerable state.
For an ideal cavity, the possibility of qubit A steerer to qubit B is shown in Fig. 5. It is clear that the degree of steerability is vulnerable for product and maximum entangled states. There are intervals of time, where the steering degree appears if the atomic system is initially prepared in the product state. The function S oscillates between its S ≃ 0.2 and zero. Meanwhile, in these intervals, the S is destroyed if the atomic system is initially prepared in the state maximum entangled stat.
In Fig. 6, we have plotted the optimal behaviour of steerability between the atomic system in the presence of the deformed field. The influence of the deformation function is similar to that illustrated in Figs. 2 and 4, where the steering degree decreases as the deformation parameter q increases. The initial setting states of the atomic system have a clear effect on the possibility of qubit A steerer to B, where for product state the function S increases as the coupling dipole increases, while this increasing rate is large if the atomic system is prepared in a maximum entangled state and a little value of deformation parameter.

Conclusion
In this paper, we investigate the influence of a quantized deformed field and atomic dipole interaction on three measures of quantum correlation quantifiers. These quantifiers are the l 1 norm of coherence, the entanglement of formation, and normalized EPR-steering, which illustrate the quantum coherence, non-separability, and steerability, respectively. This idea is clarified for a system consisting of a two-qubit interaction locally with a deformed cavity mode. It is assumed that the field is initially prepared in a coherent state, while the twoqubit is initially prepared either in a product state or a maximum entangled state. The time The temporal evolution of S against the scaled time with the same parameters as Fig. 1 evolution of the total system is obtained analytically, where the final density operator of each subsystem is obtained.
Our results show that the initial two-qubit setting state plays an important role in the general behaviour of the coherence and consequently in the non-separability and steerability. For a deformed cavity, the possibility of generating coherence, entanglement, and steering for an atomic system initially prepared in a maximum entangled state is much better than that displayed for a system prepared initially in a product state. The three quantifiers increase as the intensity of the coupling dipole decreases with a small value of the deformation rate and initial entangled state. Meanwhile, the increasing of the deformation parameter minimizes the quantum correlation for large coupling dipole and initial entangled state. A similar effect is shown for the large deformation rate and coupling dipole, where the three quantifiers decrease as the deformation and coupling dipole increase in the initial atomic product state.
For the ideal cavity, the steering is very weak, either in the product or maximum entangled states. The non-separability and coherence decrease as the intensity of the coupling dipole increase for the initial maximum entangled state. The coupling dipole slightly affects the degree of coherence and non-separability for the initial atomic product state.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). The authors have not disclosed any funding.

Availability of data and materials
The used code of this study is available from the corresponding author upon reasonable request. (c) (d) Fig. 6 The temporal evolution of S against the scaled time with the same parameters as Fig. 2