# Entanglement Control of Two-Level Atoms in Dissipative Cavities
## 3 Pancharatnam Phase and Concurrence  
We need to estimate a certain parameter for controlling the dynamics and entanglement of the system. A special attention is paid for the value of the initial latter phase parameter \(\phi\). To reach that goal, we investigate the evolution of Pancharatnam phase \(\Phi(\phi,t)=\arg(\langle\psi(\phi,0)|\psi(\phi,t)\rangle)\). To control the phase \(\phi\), we plot \(\Phi(\phi,t)\) vs. \(\phi\) for three different values of the scaled time \(\lambda t\), as in Figure 2. The red, black, and blue curves are plotted for \(\lambda t=\pi/3\), \(\pi/4\), and \(\pi/2\), respectively. In the red curve, we note that there is a smooth evolution of the phase, whereas for the black and blue curves, they exhibit two artificial phase jumps for two different values of \(\phi\). The phase jump for the blue curve (\(\lambda t=\pi/2\)) is repeated every period of \(\pi\) and in-between the jumps the evolution is semi-parabolic shaped and reflects a slow variation of the system. The two phase jumps of the black curve are repeated every \(\approx\frac{13}{20}\pi\) and in-between the two jumps the variation is very slow, smooth and is separated by \(\approx\frac{7}{20}\pi\). Now, we can detect the dynamical behavior of the considered mutipartite system, by investigating the degree of entanglement (DEM) by using the _concurrence_ measure, which was formulated as a  
Figure 2: The evolution of Pancharatnam phase \(\Phi(\phi)\). convex measure to amount the DEM for two qubits in pure states by [PERSON] [55]. For two qubits in pure states, concurrence is \(\mathcal{C}(t)=\sqrt{2\left(1-T\tau\hat{\varrho}^{2}\right)}\), where \(\hat{\varrho}=\left|\psi(t)\right\rangle\left\langle\psi(t)\right|\) is the reduced density operator. The definition of concurrence has been extended to include multiple qubits [56], and can be calculated generally by  
\[\mathcal{C}(t)=\sqrt{2\sum_{i,j=1,i\
eq j}^{4}\left(\varrho_{ii}\varrho_{jj}- \varrho_{ij}\varrho_{ji}\right)},\]  
where \(\varrho_{ij}\) are the elements of reduced density \(\varrho\) in matrix form. Figure 3 sketches the evolution of _concurrence_\(\mathcal{C}(t)\) against the scaled time \(\lambda t\). In Figure 3, we plot \(\mathcal{C}(t)\) versus the scaled time by using the estimated initial value for the latter phase \(\phi=\frac{\pi}{2}\), which is chosen due to the existence of the artificial phase jump at this value in the geometric phase in Figure 2. In Figure 3a, we set the atoms initially to be in excited (upper-most) states \(\theta=0\), and \(\varpi=\pi\lambda\), we note that DEM \(\leq\ln 2\), which is less than the standard result in models initially prepared in superposition of states. In the next figures, we examine the results of considering superposition of atomic states. In Figure 3b, we set the coupling variation parameter \(\varpi=0\), and take three various values for the decay parameter \(\gamma\) of the field. We observe that, in the beginning of the interaction between the two atoms and the coherent field \(\lambda t\leq 3\pi\), the effect of the decay parameter is not noticed and the concurrence curves are very similar, but at a drastic point of change,it differs dramatically, as we see that the black curve \(\gamma=0\), and then it fluctuates till reaching a stable case of concurrence to be \(\geq 0.8\); the red curve \(\gamma=(10^{-4}\lambda)\) has a chaotic behavior, as in the beginning. It evolves to give a higher rate of concurrence compared with the absence of decay case (black curve), and after a sufficient time it decreases. The blue curve \(\gamma=(10^{-3}\lambda)\) represents the system when concurrence rapidly reaches the disentangled state (death of entanglement). In Figure 3c, we set \(\varpi=\pi\lambda\), and we note the effect of the oscillation in the matter-field coupling as proposed in the considered model. The effect of that function is clearly noted in the higher case of the decay rate (blue curve) as the interaction has become very weak and fluctuations affect the system evolution.