\(A(t)=\frac{1}{2\hbar} \sum_{\gamma}
c_{\gamma} \{ [
\frac{\sin(\omega_{\gamma}t)}{
\omega_{\gamma}} - \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)}
- \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ]
(\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} )
\)
\(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} +
\frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1}
{2(\omega_{\gamma} - 2 \lambda)} +
\frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1}
{2 (\omega_{\gamma} + 2 \lambda) } ]
(\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \} \
\frac{\Delta_0 \epsilon}{2 \lambda^2 \hbar}
\)
\(B(t)=\frac{1}{2\hbar} \Delta_0 \sum_{\gamma}
c_{\gamma} \{ [
\frac{\cos((2\lambda - \omega_{\gamma} ) t ) - 1 }
{2 ( 2\lambda - \omega_{\gamma} ) } + \frac{\cos((2 \lambda + \omega_{\gamma} ) t ) -1 }
{2 (2\lambda + \omega_{\lambda} ) } ]
(\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} )
\)
\(\)\(\) \(+ \ i \ [ \frac{\sin((2\lambda - \omega_{\gamma} ) t ) }
{2(2\lambda - \omega_{\gamma} ) } +
\frac{\sin((2\lambda + \omega_{\gamma} ) t \ ) }
{2(2\lambda + \omega_{\gamma} ) } ]
(\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \}
\)
\(\)\(C(t)=- \frac{1}{2\hbar} \sum_{\gamma}
c_{\gamma} \{ [
\frac{\sin(\omega_{\gamma}t)}{
\omega_{\gamma}} + \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)}
+ \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ]
(\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} )
\)
\(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} +
\frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1}
{2(\omega_{\gamma} - 2 \lambda)} +
\frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1}
{2 (\omega_{\gamma} + 2 \lambda) } ]
(\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \}
\)
\(- \frac{1}{2\hbar} \sum_{\gamma}
c_{\gamma} \{ [
\frac{\sin(\omega_{\gamma}t)}{
\omega_{\gamma}} - \frac{\sin((\omega_{\gamma} -2 \lambda ) t)}{2 (\omega_{\lambda} - 2 \lambda)}
- \frac{\sin((\omega_{\gamma} + 2 \lambda) t)}{2(\omega_{\gamma} + 2 \lambda)} ]
(\hat{b}^{\dag}_{\gamma} + \hat{b}_{\gamma} )
\)
\(- \ i \ [ \frac{\cos(\omega_{\gamma} t) - 1}{\omega_{\gamma}} +
\frac{\cos((\omega_{\gamma} -2 \lambda) \ t \ ) - 1}
{2(\omega_{\gamma} - 2 \lambda)} +
\frac{\cos((\omega_{\gamma} + 2 \lambda) \ t ) - 1}
{2 (\omega_{\gamma} + 2 \lambda) } ]
(\hat{b}^{\dagger}_{\gamma} - \hat{b}_{\gamma} ) \}
\frac{(\frac{\epsilon}{\hbar} )^2 - \Delta^2_0 }
{4\lambda^2}
\)
\(\kappa=\sqrt{A(t)^2+B(t)^2+C(t)^2}.\)
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.