Effect of Classical and Quantum Superposition of Atomic States on Quantum Correlations

. In this paper, we report the e ﬀ ect of classical and quantum superposition of atomic states on quantum correlations. Coupled photon pairs generated in a ladder quantum beat laser using coherent-induced classical ﬁ eld and atomic state coherent superposition are considered. Once the quantum coherence get su ﬃ cient time, it can generate quantum correlations that include quantum discord, quantum entanglement, and quantum steering, which quickly increase with time until they get their maximum strength. Their strength can be improved further by increasing the number of superposed atoms per unit time, selecting an appropriate amplitude of the classical ﬁ elds, and managing the amount of temperatures and phase ﬂ uctuations. In particular, two-way quantum steering, which is a guarantee for the existence of quantum discord and quantum entanglement, can be achieved by increasing the rate of atomic injection from 2 kHz to 25kHz even if the mean temperature of the heat bath is considered. The maximum achievable strength of quantum correlations is enhanced by increasing the rate of atomic injection and choosing an appropriate parameters of the coherent-induced classical ﬁ eld in the open quantum system which is treated by using the density operator approach.


Introduction
Quantum correlations have been the subject of intensive studies in the past two decades, mainly due to the general belief that they are a fundamental resource for quantum information processing tasks. The first rigorous attempt to address the classification of quantum correlation has been put forward by Werner [1], who put on a firm basis on the elusive concept of quantum entanglement. Quantum superpositions are quantum mechanical principles that are at the heart of quantum information and communication applications [2]. In addition, quantum superposition, which is often called quantum coherence of two states, plays a fundamental role in quantum information and communication protocols. Quantum coherence is a local and nonclassical correlation of two or more quantum states [3,4]. Quantum coherence is responsible for quantum features such as quantum discord, quantum entanglement, and quantum steering. Quantum features produced by the interaction of atoms and photons are powerful and suitable for quantum information and communication due to the relevant physical mechanisms and sources of decoherences that can be understood and easily modeled [5][6][7][8]. In particular, quantum features generated from strongly coupled photon pairs have been a long-sought goal in quantum information and communication in view of the fact that photons combine at high speed with long coherence times at room temperature [9]. To this aim, the modelling scheme that would generate strongly coupled photon pairs has becomes a great topic of interest among the current quantum information and communication researches [10,11]. Generating and maintaining a strong quantum coherence between quantum particles, especially in the presence of decoherence, is one of the central goals of quantum information and communication protocols [12]. To this effect, researches have been conducted to study the role of thermal light on quantum entanglement generated by a cascaded quantum beat laser, and the mean photon number of thermal light appears to degrade the entanglement [13]. Ullah et al. have studied quantum features in two-photon correlated spontaneous emission lasers [14]. They have proved that a correlated spontaneous emission laser is an active quantum system with invaluable applications in quantum information processing. Originally, quantum beat lasers were used for lasing without population inversion and quenching of spontaneous emission noise [15]. Recent studies show that quantum beat laser systems are a suitable source of quantum squeezing and quantum entanglement in which the quantum coherence is induced by driven fields [16,17]. These quantum features may suggest the existence of other quantum features such as quantum discord and quantum steering, although they are not investigated so far. When correlated quantum states are used as a resource for teleportation, teleportation fidelity reveals two different aspects of nonclassicality or measures of correlations of quantum nature. The quantum correlations between the output two-mode Gaussian state of the cavity field depend on the Rabi frequency of the coupling fields and on the cavity damping rates. Specifically, the time evolution of quantum discord and entanglement become independent (dependent) of the purity (nonclassicality) of the initial states of the cavity field. Nonetheless, both the nonclassicality and purity, in the latter case, influence the quantum steering and nonlocality. Consequently, quantum nonlocality leads steerability, and quantum steering shows entanglement in the correlated field of the laser cavity. The reverse, however, does not exhibit always [18].
The main objective of these article is to study the effects of classical and quantum superposition of atomic states on quantum steering, entanglement, and discord by applying the master equation approach when the quantum coherences built up between photon pairs in a ladder quantum beat laser subjected to quantum decoherences originated by phase fluctuation and temperature of heat bath.

The Model and Dynamics of the System
We consider a three-level atomic system in cascade configuration with energy levels jai, jbi, and jci which is placed inside a doubly resonant cavity as shown in Figure 1. Initially, the atom is prepared in coherent superposition of the ground state jai and the upper excited state jci. The two modesô 1 andô 2 to be at resonance with the two transitions jci ⟶ jbi and jbi ⟶ jai are dipole allowed and direct transition between level jci and level jai to be dipole forbidden.
The quantum Hamiltonian expressions describing the interaction pictures of the quantum beat lasers have been derived using the rotating wave and electric dipole approximations and can be written as [19,20] H =Ĥ 1 +Ĥ 2 : The interaction Hamiltonian of a quantum beat laser with the annihilation operatorsô 1 andô 2 and classical field is expressed using the unitary operatorÛ = exp ð−iĤ 1 tÞ, With ξ being the classical field amplitude, here, we take the initial state of a single three-level atom considered: where ϕ is is an arbitrary phase difference between the two states. The initial density operator for a single atom is expressible in the following form: where θ represent the deviation of the phase fluctuation, ρ ð0Þ aa = jC a ð0Þj 2 and ρ ð0Þ cc = jC c ð0Þj 2 are the probability for the atom to be in the upper and lower levels at the initial time, and ρ ð0Þ ac = C a ð0ÞC * c ð0Þ and ρ ð0Þ ca = C c ð0ÞC * a ð0Þ represent the atomic state superposition at the initial time. We also note that jρ ð0Þ ac j 2 = jρ ð0Þ ca j 2 = ρ ð0Þ aa ρ ð0Þ cc . Considering the concept of the linear and adiabatic approximation schemes in the good cavity limit into account, we can find that the equation of evolution of the density operator for the cavity modes has the following form:  Figure 1: Schematic representation of atomic laser system coupled to reservoir, where r a is the rate of atomic injection into the cavity and κ (kappa) is the cavity damping constant.

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We find the solutions of the coupled differential Equations (21)-(22) as the following: in which H ± = I ± ðt + τÞ + J ± ðt + τÞ, where in which where

Quantum Steering via Coherent Classical Field
Quantum steering refers to the nonclassical correlations that can be observed between the outcomes of measurements applied on half of an entangled state and the resulting postmeasured states that are left with the other party. From an operational point of view, a steering test can be seen as an entanglement test where one of the parties performs uncharacterised measurements. Thus, quantum steering is a form of quantum inseparability that lies in between the wellknown notions of the Bell nonlocality and entanglement. It is a quantum correlation that can exist if and only if the system is entangled. Therefore, it is stronger than entanglement. Let us consider local Gaussian measurements performed on modeô 1 . A Gaussian state, b ρ = ρðô 1 ,ô 2 Þ for two-mode photons, isô 1 ⟶ô 2 steerable if the following condition is violated [18,22,23]: in which Ωô 2 is the covariance matrix pure states in the second system described by is the covariance matrix of photon pairs, and 0ô 1 indicates the null matrix in the first system.
It is a common knowledge that Equation (33) is satisfied when L 1 > 0 and ðL 2 − L T 12 L −1 1 L 12 Þ + iΩ o 2 ≥ 0. The first condition L 1 > 0 is always true since the operator representing these measurements is a positive covariance matrix. Therefore, L isô 1 ⟶ô 2 steerable only if the second condition is violated. That is, when the matrix where L can be diagonalized using symplectic transformations that yield physically interesting symplectic eigenvalues of L 2 L , which we have already derived as ðs ± Þ 2 = ða ± ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 2 − 4 det ðLÞ p Þ/2, where a = det ðL 1 Þ + det ðL 2 Þ + 2 det ðL 12 Þ and s ± are invariant quantities. In terms of these eigenvalues,ô 1 ⟶ô 2 steerability is given by Quantum steering is asymmetric since it is one-way and device-independent quantum correlation. Therefore, one could easily swap the roles of L 1 and L 2 to obtain the measurement of Gaussian quantum steering, Sô 2 ⟶ô 1 . According 4 Advances in Mathematical Physics to these formulations, Gaussian quantum steering is exhibited only when Sô 1 ⟶ô 2 > 0 and Sô 2 ⟶ô 1 > 0. However, there would be no information on the upper bounds. This may be an issue to correctly measure the degree of quantum steering. Therefore, we follow another formalism in which case, the necessary and sufficient condition for two-mode Gaussian quantum steer-ability fromô 1 ⟶ô 2 is Here, the conditional variances are defined by ½Δx 3j1 2 = ½Δðx 3 − g x x 1 Þ 2 and ½Δx 4j2 2 = ½Δðx 4 + g p x 2 Þ 2 , in which g x and g p are optimization factors which are real constants adjustable to minimize quantum fluctuations corresponding to the previously defined quadrature operators. After detailed mathematical manipulation in the optimization conditions, we arrived at with the optimization factors g x = l 13 ðtÞ/l 33 ðtÞ and g p = −l 24 ðtÞ/l 33 ðtÞ.
To investigate Sô 1 ⟶ô 2 and Sô 2 ⟶ô 1 , we plot Sô 1 ⟶ô 2 and Sô 2 ⟶ô 1 versus the dimensionless parameter ξ/γ by varying the relevant system parameters. An in-depth analysis of quantum steering and its comparison with other quantum features can be made using 2D plots as follows. Figures 2 and 3 depict the variation of the quantum steering with coherent classical fields and the parameters characterizing the system as a whole. We also notice that the quantum steering declines with increased temperature of the heat bath describing via n th and phase fluctuations. On the other hand, injecting more atoms quickly into the cavity and tuning the coherent classical field to an appropriate intensity can efficiently oppose the negative impact of quantum decoherences to generate one-way quantum steering from Sô 1 ⟶ô 2 . It is a peculiar behavior of the manifested quantum steering that disappears in the strong coupling regime, where notable quantum discord and entanglement are observed under the same system parameters. This must be attributed to the quantum coherence built up by a coherent classical field in the strong coupling regime which is not strong enough to generate quantum steering, which is one of the restrictive nature of quantum correlations.

Quantum Steering via Coherent Superposition
Dependence of one-way Gaussian quantum steering on Sô 1 ⟶ô 2 is demonstrated in Figures 4-6 under different system parameters. From these figures, Sô 1 ⟶ô 2 ≥ 1 for η = 1, in which the superposition-induced atomic coherence vanishes completely since all atoms occupy at the ground energy level with 100% probability. The maximum achievable strength of quantum steering occurs when atoms are found with 48.5% of the excited energy level, for which other quantum features are also stronger. We also observe a rapid increase of the strength of quantum steering with increased atomic injection rates. However, after attaining its peak value, quantum steering rapidly decreases with an increased probability of the atoms in the ground energy level. One can also see that the quantum steering decreases with the temperature of the heat bath and atomic phase fluctuations.

Quantum Entanglement via Coherent Classical Field
A pair of quantum mechanical particles is in an entangled state if and only if their individual states cannot be expressed as a product of the states of its separate constituents. Thus, one can write where P i ≥ 0 and ∑ i P i = 1 represent the normalization condition for the combined density state of the composite system. Although numerous criteria of entanglement measures have been developed and currently available in the literature, here, we apply the criterion set by Duan et al. [24]. This criterion is highly interesting due to its direct utilization to quantify quantum squeezing.
Based on this criterion, a quantum state of the system is entangled if the sum of the variances of the EPR-type operatorsx andp satisfies the condition: Þðô † 2 −ô 2 Þ are the quadrature operators of the cavitymode photons. Now, we find the variances ofx andp. The variancex can be obtained from the relation Δx 2 = hx 2 i − hxi 2 , where It is now not difficult to find Equation (41); it reads as follows: Moreover, it is straightforward to see that With the help of Equations (41) and (43), the variance of x becomes:

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Following the same procedure, it is possible to verify that Thus, the sum of the variances ofx andp is found to be where n 1 = ho * 1 o 1 i, n 2 = ho * 2 o 2 i, and n 12 = ho 1 o 2 i are clearly given by Equations (A.17)-(A.20). To investigate the entanglement of the two-mode radiation with this criterion, ðΔ x 2 + Δp 2 Þ/2 is plotted for the same system parameters used in the previous section. The quantum system exhibits 100% entangled and nonentangled when the sum of fluctuations in the position and momentum-like operators reduces to zero, ðΔx 2 + Δp 2 Þ/2 = 0, and greater than one, ðΔx 2 + Δ p 2 Þ/2 > 1, respectively.
The dependence of quantum entanglement on time, coherent classical field, and coherent superposition can be inferred from Figures 6 and 7. Variation of quantum entanglement with coherent classical fields under different system parameters is demonstrated in Figures 7-9, in which the quantum entanglement is lost between ξ = 3γ/4, and ξ = 6γ. However, it increases for certain values of the parameter of the coherent-induced classical field and attains peak values around ξ = γ/10 and ξ = 10γ. We show that the photon pairs are in a maximally entangled state boosted with an increased atomic injection rate. For the same system parameters, the quantum discord states for light modes are also strong. This would probably be one of the interesting results of our system for it can simultaneously support entangled state or discordant state-based quantum information processing. On the other side, the photon pairs are in a slightly changing entangled state, which could vary depending on other units of quantum system, for ξ > 10γ. However, the entanglement declines with increasing parameters of quantum decoherences heat bath and atomic phase fluctuation.

Quantum Entanglement via Coherent Superposition
The effect of atomic state superposition on the dynamics of quantum entanglement is shown in Figures 10-12. The influences of system parameters such as time, phase fluctuations,     Advances in Mathematical Physics and atomic injection rate on the dynamics of entanglement are negligible for η = 1 but considerable for η = 0. Entanglement of photon pairs is maximized near the maximum atomic coherence, η = 0:23, and enhanced with the help of r a as shown in Figure 11. Under the same condition, the entanglement of photon pairs at t = 10 s is more by 20% than that at a steady state for η = 0 and N th = 0 (see Figure 10). On the other hand, we show in Figure 12 that the entanglement of photon pairs is completely lost by small phase fluctuations of θ ≥ 0:0125. Therefore, the investigation of quantum features without considering phase fluctuations is just an assumption far from participial reality.

Quantum Discord via Coherent Classical Field
Quantum discord is defined as a measure of the quantum correlations present in a system. The difference arises because of the role played by measurement on the system. Quantum discord has been of relevant research interest during the past few decades in light of its potential as a resource of quantum information and communication, especially in disentangled quantum systems.
In the classical systems, it is possible to obtain all information of the system without disturbing it. However, this is not the case in quantum mechanics since measurements can in general modify quantum systems. Owing to this fact, the two equivalent expressions of mutual information in classical information theory are not the same for quantum systems. The difference between these equivalent quantities is used to define the quantum discord Dô 1 , which is a quantum correlation beyond entanglement, given by where b ρ = ρðô 1 ,ô 2 Þ represents the density operator of the combined system, and is the quantum mutual information that provides a measure of the total correlation within a bipartite system. In this descriptions, Sðô 1 Þ and Sðô 2 Þ represent the Von Neumann entropy of each mode while Sðb ρÞ is the joint Von Neumann   Advances in Mathematical Physics entropy of the bipartite system. On the other hand, Jðb ρÞ represents the classical mutual information that captures the classical correlations given by [25] Jðb ρÞ = Sðô 1 Þ − Sðb ρ Here, the term Sðb ρðô 1 ,ô 2 Þj Q ĵ o 2 Þ indicates that the conditional Von Neumann entropy of modesô 1 andô 2 , with Q ĵ o 2 = jjô 2 ihjô 2 j being a set of local projective measurements on light modeô 2 . An optimization condition should be imposed on Jðb ρÞ to avoid the dependence of quantum discord on the measurements [26,27]. This reads as follows: The optimization process of measurement of mutual information highly complicates the quantification of quantum discord in general. Analytical formulas of quantum discord are derived for continuous variable Gaussian bipartite systems in Refs. [28,29]. Continuous variable Gaussian states have Gaussian Wigner functions. It is well known that a bipartite Gaussian system is fully described by its symplectic eigenvalues, covariance matrix, and the standard form of diagonal subblocks. This reads as follows: in which L 1 and L 2 are the covariance matrices describing each mode separately, while L 12 are the intermodal correlations. The elements of the matrix in Equation (27) can be obtained from the relation [29] in which i, j = 1, 2, 3, 4. The quadrature operators are defined asx 1 =ô 1 +ô † 1 , Using these definitions, the only nonvanishing terms in the extended covariance matrix are where l 11 ðtÞ = l 22 ðtÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi det ðL 1 Þ p = 2n 1 ðtÞ + 1, l 33 ðtÞ = l 44 ðtÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi det ðL 2 Þ p = 2n 2 ðtÞ + 1, a n d l 13 ðtÞ = −l 24 ðtÞ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi −det ðL 12 Þ p = 2n 12 ðtÞ. One can also easily verify that On the other hand, the corresponding eigenvalues of the symplectic matrix described in Equation (51) are given by where a = det ðL 1 Þ + det ðL 2 Þ + 2 det ðL 12 Þ and s ± are invariant quantities. The continuous variable Gaussian quantum discord for two-mode radiation described by a covariance matrix M is given by where

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Advances in Mathematical Physics with c = ½ðl 11 Þ 2 + 1½det ðLÞ + ðl 33 Þ 2 − ½det ðLÞ − ðl 11 l 33 Þ 2 2 , and we also note that for any variable x, the function IðxÞ is defined by IðxÞ = ðð1 + xÞ/2Þ log 2 ðð1 + xÞ/2Þ − ðð1 − xÞ/2Þ log 2 ðð1 − xÞ/2Þ. In these equations, we have not shown time dependence to save environment. The quantum discord for modeô 2 can be easily obtained by swapping the role of l 11 ðtÞ and l 33 ðtÞ. This reads as follows: where According to the definition of quantum discord, twomode radiation fields are entangled for Dô 1 ðDô 2 Þ > 1, and the two-mode fields can either be in a separable or entangled state for 0 ≤ Dô 1 ðDô 2 Þ < 1. The effect of coherent classical field interaction on atomic transitions from the excited state and ground state is clearly demonstrated in Figures 13-15. We selected a specific value of system parameters based on the previously realized experimental schemes for microwave setup [24,30] and recognized the theoretical articles [31]. It is clearly indicated that the quantum discord is stronger in the absence of the coherent classical fields, ξ = 0. For ξ ≠ 0, the classical field begins contributing its part. Quantum discord for modeô 1 sharply declines with increasing classical field parameters until ξ ≈ γ, then its slowly increases to attain a certain maximum value around at ξ = 3:5γ, and later, it slowly decreases to zero depending on the values of other parameters such as r a and N th . This may occur because the classical field does not induce quantum coherence uniformly, and hence, an appropriate value of its parameter can be chosen to get an optimized quantum discord. This quantum feature can highly be increased by adding temperature of the heat bath and increasing the atomic injection rate (see Figures 13 and 14). Moreover, increasing phase fluctuations could slightly improve the quantum discord as can be seen in Figure 15.

Quantum Discord via Coherent Superposition
Dependence of quantum discord for modeô 1 on the quantum coherence induced by quantum superposition of atomic states has been shown in Figures 16-18 under different system parameters. Analytical calculations have indicated that the quantum coherence induced by the atomic state superposition is maximum for η = 0 and minimum for η = 1 (see Equations (19) and (20)). These figures also manifest the same features: quantum discord becomes weak and strong when the quantum coherence induced by atomic state superposition is minimum and maximum, respectively. This must happen due to the fact that quantum coherences are a resource for quantum features including quantum discord.

Conclusion
In this paper, we have studied the effect of classical and quantum superposition of atomic states on quantum correlations. The photon pairs were strongly coupled through coherent-induced classical fields and atomic state superposition and were subjected to two different kinds of quantum decoherence due to coupling with heat bath and atomic phase fluctuations. The dynamics of quantum correlations 11 Advances in Mathematical Physics

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Advances in Mathematical Physics have been observed and analyzed using the c-number master equations by taking into account different system parameters. The maximum achievable strength of quantum correlation is enhanced by increasing the rate of atomic injection, choosing an appropriate parameter of the coherent-induced classical fields which can adjust the atomic populations to the desired energy levels. The effect of atomic injection rate advances as time goes on and is more effective when 48.5% and 51.5% of atoms populate in the excited and ground energy levels. This result is attributed to the fact that atoms require sufficient time to interact and transfer their coherence to the photon pairs. On the other hand, it has been observed that the classical field is efficiently used up to induce quantum coherence only when sufficient numbers of atoms are available in the excited energy state while pumping the atoms from the ground state to the excited state would be an additional task for it when none of them are initially in the excited state. This is why quantum steering disappears in the strong coupling regime in which notable quantum discord and entanglement are observed under the same system parameters. Perceptible from the quantum discord that increases with quantum decoherence, the quantum entanglement and steering wipe out with increased decoherences originated from the temperature of the heat bath and atomic phase fluctuations. The strength of quantum steering stays for a long period of time under thermal decoherence but slightly declines with time under phase fluctuations. Therefore, the investigation of quantum features without considering quantum decoherences is just an assumption far from participial reality. The quantum scheme would probably be one of the best optical devices to simultaneously support steerable states, entangled state, and discordant state-based quantum information processing. 18Þ

Data Availability
The data used to support the findings of this study are included within the article.

Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.