c_2$ to those from Kien et.al.[@Kien-LKOHS-1999pra] (KLKOHS) and Hickman et.al. [@Hickman-PB-1986pra] (HPB). This is useful as a fixed point of refernce beween the approaches, enabling quick conversions between the parameter variables.

$$\begin{aligned}
\textrm{cf. (KLKOHS) eqn.(13): } ~~~~ ~~~~
\frac{d}{dt}
\left[
\begin {array}{c}
c_a
\\
c_b
\end {array}
\right]
&=&
 -\frac{1}
       {\imath \hbar}
\left[
\begin {array}{cc}
\hbar \Omega_{aa} & \hbar \Omega_{ab} \\
\noalign{\medskip}
\hbar \Omega_{ba} & \hbar \Omega_{bb} - \delta \\
\end {array}
\right]
~
\left[
\begin {array}{c}
c_a
\\
c_b
\end {array}
\right]
,
\label{eqn-KLKOHS-dcdt}
\\
\textrm{where $B_q$ are KLKOHS's $E_q$ envelopes:}
~~~~ ~~~~
  E 
&=&
  \frac{1}{2} 
  \left[
     B_q e^{\imath \Xi} + B_q^* e^{-\imath \Xi} 
  \right]
\\
  \hbar \Omega_{aa}
&=&
  \frac{\hbar }{2}
  \sum_q
    a_q B_q B_q^*
\\
  \hbar \Omega_{ab}
&=&
  \frac{\hbar }{2}
  \sum_q
    d_q B_q B_q^*
~~~~ ~~~~
=
 \hbar \Omega_{ba}^*
\\
  \hbar \Omega_{bb}
&=&
  \frac{\hbar }{2}
  \sum_q
    b_q B_q B_{q+1}^* \end{aligned}$$

$$\begin{aligned}
\textrm{cf. (HPB) eqn.(13): } ~~~~ ~~~~
  \frac{d}{dt}
  \left[
   \begin {array}{c}
      c_1
      \\
      c_2
    \end {array}
  \right]
&=&
  -\frac{1}{\imath \hbar}
  \left[
    \begin {array}{cc}
      -H_{11} & -H_{12}  \\
      \noalign{\medskip}
      -H_{21} & -H_{22}  \\
    \end {array}
  \right]
~
\left[
\begin {array}{c}
c_1
\\
c_2
\end {array}
\right]
\label{eqn-HPB-dcdt}
,
\\
\textrm{where $\alpha_{Hij}$ are HPB's $\alpha_{ij}$ parameters:}
~~~~ ~~~~
  H_{11} 
&=&
  