ahead to the definition of $f=
\bar{\alpha}_{12}/2\hbar =  \left( \alpha_{12}+\alpha_{21} \right) /2\hbar$ gives us $\Omega_{ab}=f \mathscr{I}$, noting $\mathscr{I}=2 A_q A_{q+1}^*$, compared to $\mathscr{I}=\frac{1}{2}E_q E_{q+1}^*$. Note that KLKOHS have defined $\Omega_{ab}=\frac{1}{2} \sum_q d_q E_q E_{q+1}^*$, so that $f=d_q$.

Two photon Bloch equations (cont)
---------------------------------

An important difference between KLKOHS & HPB, and my equations is that my $\alpha_{ij}$ coupling parameters retain a “slow” time dependence at the Raman frequency $W_{12}$. I now turn my equations (\[eqn-single-dcdt\]) into Bloch equations by first working out $d/dt$ of $c_1^* c_1$, $c_2^* c_2$, and $c_1^* c_2$,   $$\begin{aligned}
 \imath \hbar
\frac{d c_1^* c_1}{dt}
=
 \imath \hbar
c_1^* \frac{d c_1}{dt}
+
 \imath \hbar
c_1 \frac{d c_1^*}{dt}
&=&
- \alpha_{11}   \mathscr{I} c_1^* c_1 
- \alpha_{12}   \mathscr{I} c_1^* c_2
+ \alpha_{11}^* \mathscr{I} c_1 c_1^*
+ \alpha_{12}^* \mathscr{I} c_1 c_2^*
\\
&=&
+ \alpha_{12}^* \mathscr{I} c_1 c_2^*
- \alpha_{12}   \mathscr{I} c_1^* c_2
- \left[ \alpha_{11} - \alpha_{11}^* \right] \mathscr{I} c_1^* c_1
\\
&=&
+ \mathscr{I} \left( \alpha_{12}^* c_1 c_2^* - \alpha_{12} c_1^* c_2 \right)
; ~~~~ ~~~~ ~~~~ ~~~~ \textrm{since} ~~ \alpha_{11}=\alpha_{11}^*
\\
 \imath \hbar
\frac{d c_2^* c_2}{dt}
=
 \imath \hbar
 c_2^* \frac{d c_2}{dt}
+
 \imath \hbar
 c_2 \frac{d c_2^*}{dt}
&=&
- \alpha_{21} \mathscr{I} c_2^* c_1 
- \alpha_{22} \mathscr{I} c_2^* 