the real quantum optical and quantum dissipative systems, the counterrotating terms involving higher excited states, $%
a_{k}^{\dag }\sigma _{+}$ and $a_{k}\sigma _{-}$, can be neglected, a condition which is the so-called rotating-wave approximation (RWA); thus the full Hamiltonian (\[spinboson\]) can be reduced to the following RWA form$$H_{SB}^{RWA}=\frac{\Delta }{2}\sigma _{z}+\sum_{k}\omega _{k}a_{k}^{\dag
}a_{k}+\sum_{k}g_{k}\left( a_{k}^{\dag }\sigma _{-}+a_{k}\sigma _{+}\right) .
\label{RWA}$$It is generally believed that the RWA is a reasonably good approximation because the counterrotating terms violate energy conservation, leading to virtual processes, and thus are suppressed.

The total excitation number of the spin-boson model is $\hat{N}%
=\sum_{k}a_{k}^{\dag }a_{k}+\sigma _{+}\sigma _{-}$, where $\sigma _{\pm
}=\left( \sigma _{x}\pm i\sigma _{y}\right) /2$. With (without) the RWA, the system possesses a $U(1)$ ($Z_{2}$) symmetry, which is characterized by the action of the operator $\Re \left( \theta \right) =\exp \left( i\theta \hat{N%
}\right) $, where the arbitrary angle $\theta $ corresponds to $U(1)$ symmetry and the special value of $\theta =\pi \ $  corresponds to $Z_{2}$ (parity) symmetry. The parity operator $\hat{\Pi}=\Re \left( \pi \right) $ has two eigenvalues  $\pm 1$. With the advance of modern technology, various qubit and oscillator coupling systems can be engineered in many solid-state devices, such as superconducting circuits [@Niemczyk; @Yoshihara], cold atoms 