value of collective spin operators can be expressed by $P_m$, such as $4\langle S_{y}^{2}\rangle =4\sqrt{3}({}_{-\frac{3}{2%
}}\langle n-2|n\rangle _{\frac{1}{2}}P_{-\frac{3}{2}}P_{\frac{1}{2}}+{}_{-%
\frac{1}{2}}\langle n-1|n+1\rangle _{\frac{3}{2}}P_{-\frac{1}{2}}P_{\frac{3}{%
2}})-4(P_{-\frac{1}{2}}^{2}+P_{\frac{1}{2}}^{2})+3$, we calculate the concurrence $C$ by the zeroth-order approximation and the GRWA, respectively.

![(Color online) Dynamics of the concurrence for the qubit-qubit entanglement with the initial W state for the ultrastrong coupling strength $%
g/\protect\omega=0.1$. The parameters are the same as in Fig. [dynamics GEM]{}.[]{data-label="dynamics concurrence"}](condyn.eps)

We plot the dynamics of the concurrence for different detunings $\Delta /\omega =0.1$ and $1$ in Fig. \[dynamics concurrence\]. The initial W state gives the maximum pairwise entanglement $C=2/3$ of any Dicke states. Fig. \[dynamics concurrence\] (a) shows that dynamics of the concurrence by the zeroth-order approximation are similar to the numerical ones in the off-resonance case $%
\Delta /\omega =0.1$, in which the RWA results are invalid. The sudden death of the bipartite entanglement is observed in the resonance case in Fig. [dynamics concurrence]{} (b). The dynamics of the concurrence obtained by the GRWA is similar to the numerical results, exhibiting the disappearance of the entanglement for a period of time. However, there is no sudden death of the entanglement in the RWA case, indicating that the CRW terms are not 