Influence of the dissipation on the N-level atom interacting with a two two-level atoms in presence of qubit–qubit interaction

The interacting of two qubits and an N-level atom based on su(2) Lie algebra in the presence of both qubit–qubit interaction and dissipation term is considered. The effects of the qubit–qubit interaction and the dissipation term on the dynamics of the proposed system are discussed in detail for certain values of the number of levels. The dynamical expressions of the observable operators are obtained using the Heisenberg equation of motion. The population inversion and linear entropy, as well as the concurrence formula as a measure of entanglement between the two qubits are calculated and discussed. The roles of the number of levels, the qubit–qubit coupling parameter and the dissipation rate on these quantities are also discussed. We explore the sudden birth and sudden death of the entanglement phenomena with and without the dissipation term.


Description of the model and solution
An application of the su(2) algebraic system has been presented in 31,32 . They examined the star network of spins, where all rotations interact exclusively and continuously with a central rotation through Heisenberg XX couplings of equal strength. They considered that the spin star as central qubit interacts with all other qubits, and the non-central qubits do not interact directly with each other. Therefore, the quantum system that represents this phenomenon is given as follows, The parameters ω and k 1 , k 2 are respectively the frequencies of the central qubit and the surrounding ones, while is the coupling parameter, Rrepresents the coupling between qubits themselves. The Ŝ k kl and π j z denote the generators of operators of su (2) which are obey the commutation relation: We add the dissipation term and use the collective operator Ĵ k = N j=1 π j k , (k = z, +, −). Therefore, equation (1) becomes, where γ denotes the decay rate, Ĵ − ,Ĵ + denote the lowering and raising operators of the su(2) symmetry and satisfy the following commutation relation: For describes and explains the physical phenomena for the model (2 ), we write the differential equations by applying the Heisenberg equations of motion (HEM). Therefore, the mathematical expressions of the HEM take the form from which we can show that, 12 ,

12Ŝ
(2) Under the exact resonance case condition (δ 1 z +δ 2 z = 0) and for space of the two qubits as |ee� , |eg� , |ge� and |gg� the analytical form of the time evolution operator Û (t) is given by where Now we assume that the initial conditions for the present system (2 ) is set as follows, the two qubits in their the excited states, while the N-level atom starts from the atomic coherent state, which is given as follows, where, |β� can be expressed as the following, the coupling coefficients Q j n (�, �) is obey the following relation, where C 2j j+n is the binomial coefficient, � ∈ [0, π] and � ∈ [0, 2π] . The exact solution |�(t)� for t > 0 is The functions Y 1 , Y 2 , Y 3 and Y 4 are given by www.nature.com/scientificreports/ After obtained on the wave function (15) can be used to describe the evolution of some quantum quantifiers as: population inversion, linear entropy and concurrence.

Population inversion
The population inversion is one of the important quantities in quantum information. Based on its results, we can determine the periods of collapse and revival, which are useful in determining the periods of maximally entangled state and purity periods (separable state) 33,34 . The population inversion function is identified as the difference between the probability of finding the qubit in the excited and the ground states. When the qubit begins in the excited states, the population inversion takes the form of First, we consider the two qubits in their excited states initially and exclude both of the qubit-qubit interaction ( R = 0 ) and the dissipation term ( γ = 0 ). In the case of j = 2 , the population inversion ranges randomly from 0.5 to 1. That is, the population inversion almost occurs in the excited state. Additionally, the phenomena of collapse and revival does not appear. After increasing the number of levels to 40, the W(t) function fluctuates between −0.5 to 0.5. Moreover, the previous random form becomes somewhat regular and periodic. A single period of collapses and revivals appears clearly, as shown in Fig. 1a. When the dissipation of the interaction cavity is considered, the amplitude of the oscillations has gradually decreases until reaching the stationary state. The results confirm that the dissipation effect is large in the case of a large number of levels, see Fig. 1b. When combining qubit-qubit interaction ( R = 5 ) and neglecting dissipation ( γ = 0 ), in the case of a small number of levels, the lower values rise up and the amplitude of the oscillations decrease compared to the previous case. When the levels increase to 40, the period of collapse decreases, and the amplitudes of the vibrations decrease, as seen in Fig. 1c. When adding dissipation to the interaction, the function W(t) decreases slowly until it reaches a state of stability after a time longer than the previous case in (Fig. 1b). Moreover, the effect of dissipation resembles the previous case when the case of the number of levels is large, see Fig. 1d.

Linear entropy
Entanglement has a pivotal role in quantum information (QI). However, it is a weak phenomenon in real experimental conditions because the quantum system is affected by the surrounding environment. These external influences lead to the disentanglement of parts of the system and eventually produce mixed quantum states 35 . The experimental and theoretical studies of quantum optical systems give a deep look at the links between mixedness and entanglement. The mixedness associated with a decrease in the quantum state purity is measured by linear entropy 36 , which is given by www.nature.com/scientificreports/ www.nature.com/scientificreports/ Here we take the same conditions as the previous section to explore the linear entropy. When the number of levels is small, the dissipation and qubit-qubit interaction are neglected. Linear entropy has periodic oscillations and oscillates between 0 and 0.3. The P(t) function reflects weak entanglement and approaches zero regularly almost every (5.5)n, n = 0, 1, 2, ... as seen from Fig. 2a. When the number of levels is large, the maximum and minimum values for linear entropy move upward. This reflects a strong entanglement between the su(2) system and the qubits. Moreover, the minimum values achieved at the beginning of the collapse period and the center of the revival period. The maximum values achieved at the beginning and end of the revival period, see Figs. 1a,2a.
After takes the dissipation parameter into account, the P(t) function stabilizes to its maximum value. It is indicated that, when the number of levels is small, the P(t) function needs a long time to reach the stationary state, as shown in the Fig. 2b. When the qubit-qubit interaction is taken into account and dissipation is excluded, with regard to the small number of levels, the amplitude of previous fluctuations decreases and the linear entropy reaches the smallest values periodically. This phenomenon is repeated almost every 10n, n = 0, 1, 2, .. . Whereas in the case of a large number of levels, the effect of the qubit-qubit interaction is very weak, see Fig. 2c. After combining the dissipation term, the oscillations fade quickly when the number of levels is large, while they fade slowly when the number of levels is small. The P(t) function needs more time to reach stability, as seen in Fig. 2d.

Concurrence
The entanglement for bipartite systems has been represented by many measures. One of these measures is concurrence, which represents the approximate number of Bell pairs required to prepare a state using only local canonical transformations and classical communication. In two-qubit systems, the fully closed expression of concurrence is obtained by 37 www.nature.com/scientificreports/ where a k (k = 1, 2, 3, 4) are the eigenvalues of the square roots of the density matrix R = ρ AB (σ y ⊗ σ y )ρ * AB (σ y ⊗ σ y ) and σ y is the Pauli matrix. ρ * AB is the complex conjugate of ρ AB . Concurrence vanishes, , namely C Q (t) = 0 , for unentangled qubits, whereas C Q (t) = 1 , for maximally entangled qubits.
With the same preconditions, we are now discussing the concurrence. We neglect the dissipation term and qubit-qubit interaction. When the number of levels is small, the concurrence fluctuates regularly between 0 and 0.4. In addition the phenomenon of sudden birth and sudden death are achieved at many points during the considered time. When the number of levels increases, the concurrence behavior changes completely, as the previous regular shape becomes random. The maximum value realized at the beginning of the collapse period and at the midpoint of the revival. The minimum values are realized at the beginning and end of the revival period, as shown in the Fig. 3a. It is indicated that, the values of the extreme values of the C Q (t) function are inversely proportional to the linear entropy by comparison of the behavior of the curves, as shown in Figs. 2a,3a. After taking the dissipation term into consideration, the oscillations of C Q (t) are quickly collapse and reach the minimum values in the case of a large number of levels, while the oscillations of the C Q (t) function decrease very slowly and need a long time until they reach the zero value. After the qubit-qubit interaction is included and dissipation term excluded, the amplitude of the vibrations decreases when the number of levels is small. The smallest values increase in the case of the number of levels is large and the phenomenon of sudden death and sudden birth almost disappears, as seen from Fig. 3c. After adding the dissipation, the C Q (t) function has the same behavior as before in the case of a large number of levels. While the amplitude of oscillations gradually decreases, but it needs a shorter time to reach the value of zero by comparing between Fig. 3b, d.

Conclusion
An open system containing the interaction of the two qubits and a N-level atom is handled in this article. The multi-level atom is represented by su(2) algebra operator. In addition, the qubit-qubit interaction is considered. The constant of the motion is obtained, the matrix of time evolution operator is calculated; thus, we obtained the wave function depending on time explicitly. Through the wave function, the phenomena of collapse and revival are studied, the entanglement between the multi-level atom and the qubits is discussed using the linear entropy. In addition to discussing the entanglement of the two qubits together by the concurrence. Through the results in the case of a closed system ( γ = 0 ), it is found that the phenomenon of collapses and revivals did not appear when the number of levels is small, and the behavior of the population inversion is chaotic. In contrast, when the number of levels is large, one period of collapse appeared, followed by a period of revival, then there are random fluctuations. In the case of the open system ( γ = 0.01 ), the oscillations have more quickly decayed when having a large number of levels than when having a small number of levels. The qubit-qubit interaction resisted the decay of oscillations slightly and impeding the arrival of population inversion to a state of stability. The entanglement between the multi-level atom and the qubits increased in the case of a large number of levels and never reached the pure state. Adding the qubit-qubit interaction in the case of dissipation increased the time of the linear entropy and the concurrence to the state of stability. Moreover, a contrast in the behavior between the linear entropy and the concurrence without and with the dissipation term.