calculate $c_1, c_2$; however we still retain the sum over all $i$ intermediates states, as they affect the coupling between $1$ and $2$. The diagonal couplings $\{ \alpha_{nj}, n = j
\}$ are real; but the off-diagonal couplings $\{ \alpha_{nj}, n \neq j \}$ undergo complex oscillations according to the difference in their energy levels.

Their [*frequency dependence*]{} is discussed after the following subsection; HPB’s corresponding parameters, which [*do not oscillate*]{}, were assumed to be frequency independent.

### CASE (ii): Electric Field Envelope {#sss-single-Eenvelope}

This strand replaces the electric field $E$ with a carrier-envelope description, but, unlike HPB, I use only a single carrier-envelope component rather than a set indexed by some integer $j$. This is necessary, because in the previous strand I ended up with coupling constants $\alpha_{nj}$ with strong frequency dependences.

I introduce the envelope and carrier [@Gabor-1946jiee] for the field:   $$\begin{aligned}
E &=& 
  \left[
    A e^{-\imath \omega_0 t} + A^* e^{+\imath \omega_0 t} 
  \right]
\label{eqn-single-EfromA}
\\
\textrm{so that } ~~~~ ~~~~
\imath \hbar \frac{d c_n}{dt}
&=&
-
\sum_l
  c_l
  d_{nl} 
  \exp \left[ -\imath \left( W_l - W_n \right) t \right]
  \left[
    A e^{-\imath \omega_0 t} + A^* e^{+\imath \omega_0 t} 
  \right]
\\
\textrm{now use~}l\rightarrow i; \textrm{~and assume 