holds; remember that it is difficult to maintain the accuracy of the approximations relied on above for the $E^2$ picture (in \[sss-single-Efield\]). I also drop the prime on the $\alpha_{nj}'$ parameters used in the (Case Iii)) “$2\left| A \right|^2$” electric field envelope model.

Since we are only interested in $c_1, c_2$, and because we keep only stationary or slowly varying terms, we can write equations for $c_1, c_2$ as –   $$\begin{aligned}
\imath \hbar
\frac{d c_1}{dt}
&=&
- \alpha_{11} \mathscr{I} c_1 
- \alpha_{12} \mathscr{I} c_2
\\
\Longrightarrow
~~~~ ~~~~
\imath \hbar
\frac{d c_1^*}{dt}
&=&
+ \alpha_{11}^* \mathscr{I} c_1^* 
+ \alpha_{12}^* \mathscr{I} c_2^*
\\
\imath \hbar
\frac{d c_2}{dt}
&=&
- \alpha_{21} \mathscr{I} c_1
- \alpha_{22} \mathscr{I} c_2
\\
\Longrightarrow
~~~~ ~~~~
\imath \hbar
\frac{d c_2^*}{dt}
&=&
+ \alpha_{21}^* \mathscr{I} c_1^*
+ \alpha_{22}^* \mathscr{I} c_2^*
\\
\textrm{In matrix form: } 
~~~~ ~~~~
\frac{d}{dt}
\left[
\begin {array}{c}
c_1
\\
c_2
\end {array}
\right]
&=&
  -\frac{1}{\imath \hbar}
 \left[
   \begin {array}{cc}
     \alpha_{11} \mathscr{I} & \alpha_{12} \mathscr{I} \\
     \noalign{\medskip}
     \alpha_{21} \mathscr{I} & \alpha_{22} \mathscr{I} \\
   \end {array}
  \right]
~
\left[
\begin {array}{c}
c_1
\\
c_2
\end {array}
\right]
\label{eqn-single-dcdt}
,\end{aligned}$$

Alternative formulations: Kien et.al. (KLKOHS), Hickman et.al. (HPB)
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This interlude compares my matrix equation for $c_1, 