expression of eigenvalues and eigenstates. In Sec.III, we discuss dynamics of the GME for the multi-qubit entanglement and the concurrence for the qubit-qubit entanglement by our method. Finally, a brief summary is given in Sec. IV.

An analytical treatment to the three-qubit cavity system
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The Hamiltonian of the three-qubit Dicke model, which describes three identical qubits coupled to a common harmonic cavity, is written as $\left(
\hbar =1\right) $ $$H=-\Delta J_{z}+\omega a^{\dagger }a+\frac{g}{2}(a^{\dagger
}+a)(J_{+}+J_{-}),  \label{ham1}$$where $a$ and $a^{\dagger }$ are, respectively, the annihilation and creation operators of the harmonic cavity with frequency $\omega $, $%
J_{i}\left( i=z,\pm \right) \ $ is the angular momentum operator, describing the three qubits of level-splitting $\Delta $ in terms of a pseudospin of length $J=3/2$, and $g$ denotes the collective qubit-cavity coupling strength.

In the RWA, the CRW terms $a^{\dagger }J_{+}$ and $aJ_{-}$ are neglected, and the Hamiltonian becomes $$H_{\mathtt{RWA}}=-\Delta J_{z}+\omega a^{\dagger }a+\frac{g}{2}(a^{\dagger
}J_{-}+aJ_{+}),$$ which is restricted to relatively weak-coupling strength $g\ll \omega $, and to the qubit-cavity near resonance, $\Delta \approx \omega $. Now, the interaction couples only $|-\frac{3}{2}\rangle |n+2\rangle $, $|-\frac{1}{2}%
\rangle |n+1\rangle $, $|\frac{1}{2}\rangle |n\rangle $, and $|\frac{3}{2}%
\rangle |n-1\rangle $ for each $n$, which are the eigenstates of the 