\(A(t)=\frac{1}{2\hbar} \sum_{\gamma} c_{\gamma} \{ [ \frac{\sin(\omega_{\gamma}t)}{ \omega_{\gamma}} - \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)} - \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ] (\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} ) \)
             \(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} + \frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1} {2(\omega_{\gamma} - 2 \lambda)} + \frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1} {2 (\omega_{\gamma} + 2 \lambda) } ] (\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \} \ \frac{\Delta_0 \epsilon}{2 \lambda^2 \hbar} \)
\(B(t)=\frac{1}{2\hbar} \Delta_0 \sum_{\gamma} c_{\gamma} \{ [ \frac{\cos((2\lambda - \omega_{\gamma} ) t ) - 1 } {2 ( 2\lambda - \omega_{\gamma} ) } + \frac{\cos((2 \lambda + \omega_{\gamma} ) t ) -1 } {2 (2\lambda + \omega_{\lambda} ) } ] (\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} ) \)
\(\)\(\)             \(+ \ i \ [ \frac{\sin((2\lambda - \omega_{\gamma} ) t ) } {2(2\lambda - \omega_{\gamma} ) } + \frac{\sin((2\lambda + \omega_{\gamma} ) t \ ) } {2(2\lambda + \omega_{\gamma} ) } ] (\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \} \)
\(\)\(C(t)=- \frac{1}{2\hbar} \sum_{\gamma} c_{\gamma} \{ [ \frac{\sin(\omega_{\gamma}t)}{ \omega_{\gamma}} + \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)} + \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ] (\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} ) \)
                  \(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} + \frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1} {2(\omega_{\gamma} - 2 \lambda)} + \frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1} {2 (\omega_{\gamma} + 2 \lambda) } ] (\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \} \)
                \(- \frac{1}{2\hbar} \sum_{\gamma} c_{\gamma} \{ [ \frac{\sin(\omega_{\gamma}t)}{ \omega_{\gamma}} - \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)} - \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ] (\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} ) \)
                  \(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} + \frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1} {2(\omega_{\gamma} - 2 \lambda)} + \frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1} {2 (\omega_{\gamma} + 2 \lambda) } ] (\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \} \frac{(\frac{\epsilon}{\hbar} )^2 - \Delta^2_0 } {4\lambda^2} \) 
\(\kappa=\sqrt{A(t)^2+B(t)^2+C(t)^2}.\)
In the weak coupling limit by taking \(c_\gamma \rightarrow 0\), the equations (A7) and (A8) are with respect to the equations (A4) and (A5), while setting \(E_{av} = 0\). Here \(c_0\) and \(c_1\) are complex numbers and are temperature dependent. \(\Psi_S(t)\) satisfies the superposition state of qubit.