-\frac{1}{4}
 \sum_j \alpha_{H11} (\omega_j) V_j V_j^*
\\
  H_{12} 
&=&
  -\frac{1}{4}
 \sum_j \alpha_{H12} (\omega_j) V_j V_{j-1}^*
~~~~ ~~~~ = 
  H_{21}^*
\\
  H_{22} 
&=&
  -\frac{1}{4}
 \sum_j \alpha_{H22} (\omega_j) V_j V_j^*\end{aligned}$$

Thus comparing the Rabi-like parts of my eqn (\[eqn-single-dcdt\]) with that of KLKOHS (my eqn (\[eqn-KLKOHS-dcdt\])) and that of HPB (my eqn (\[eqn-HPB-dcdt\])), gives   $$\begin{aligned}
  \alpha_{12} \mathscr{I} = \alpha_{12} . 2 A^*A = \hbar f' . 2 A^*A
~~~~ ~~~~
&=&
  \frac{\hbar}{2}
  \sum_q
    d_q B_q B_{q+1}^*
=
  \hbar
  \sum_q
    d_q 
    .
    \frac{1}{2}
    B_q B_{q+1}^*
~~~~ ~~~~
= 
 -\frac{1}{4}
  \sum_j
    \alpha_{H12}(\omega_j) V_j V_{j-1}^*\end{aligned}$$

   This uses KLKOHS: $ E = \frac{1}{2} \left( B_1 + B_1^* + B_2 + B_2^* \right)$\
$\rightarrow E^2 = \frac{1}{4} \left( B_1^2 + B_1^{*2} + B_2^2 + B_2^{2*} 
+ 2 B_1 B_1^* + 2 B_2 B_2^* + 2 B_1 B_2^* + 2 B_1^* B_2 
\right)$\
$\rightarrow E_{1-2}^2 \sim \frac{1}{4} 2 B_1 B_2^* = \frac{1}{2} B_1 B_2^*$\
which tells us the size of the field contribution to the KLKOHS Rabi-like term.

   This uses HPB: which tells us the size of the field contribution to the HPB Rabi-like term. (cf KLKOHS). So $\Omega_{aa}=\alpha_{11}
\mathscr{I} / \hbar$, $\Omega_{ab}=\alpha_{12} \mathscr{I} / \hbar$, $\Omega_{bb}= \alpha_{22} \mathscr{I} / \hbar$ (temporarily ignoring their detuning $\delta$); and looking 