to the rotation around the $%
y$ axis by the angle $\pi/2$, the initial Dicke state can be written as $$|D_{3}\rangle =\frac{1}{\sqrt{8}}(-\sqrt{3}|-\frac{3}{2}\rangle -|-\frac{1}{2%
}\rangle +|\frac{1}{2}\rangle +\sqrt{3}|\frac{3}{2}\rangle),
\label{initial state1}$$and the initial cavity state is the vacuum state $|0\rangle $. Based on the eigenstates $\left\{ |\varphi _{k,n}\rangle\right\} $ and eigenvalues $%
\left\{ E_{n}^{k}\right\} $ in the GRWA and the zeroth-order approximation, the wavefunction evolves from the initial state as $|\phi (t)\rangle
=\sum_{n,k}e^{-iE_{n}^{k}t}|\varphi _{k,n}\rangle \langle \varphi
_{k,n}|D_{3}\rangle $. And the three-qubit reduced state $\rho (t)$ can be given by tracing out the cavity degrees of freedom $$\rho (t)=\texttt{Tr}_{\mathtt{cavity}%
}(|\phi (t)\rangle \langle \phi (t)|).$$ We then calculate the absolute value of the minimum $E(\rho )$ to detect the GME by solving the minimum in Eq.( \[minm\]).

![(Color online) Dynamics of the GME for three-qubit entanglement with the initial W state for the ultrastrong-coupling strength $g/\protect%
\omega=0.1$ with the different detuning $\Delta/\protect\omega=0.1$ (a) and $%
\Delta/\protect\omega=1$ (b) by the GRWA method (dash-dotted lines), numerical method (solid lines), RWA (short-dotted lines), and the zeroth-order approximation (dashed lines).[]{data-label="dynamics GEM"}](GMEDym.eps)

Fig. \[dynamics GEM\] shows the $E(\rho )$ plotted against parameter $\Delta t/(2\pi)$ for different detunings $\Delta/\omega$ for the ultra-strong-coupling strength $g/\omega =0.1$. For comparison, results from numerical exact 