\sum_j 
    c_j 
    e^{-\imath \left( W_j - W_n \right) t}
    \left< n \right|
      d.E
    \left| j \right>
.\end{aligned}$$

We now use perturbation theory, & $d_{nm} = \left< n \right| \hat{d} \left| n \right>$, following two independent (but related) strands.

### CASE (i): Electric field {#sss-single-Efield}

This strand leaves the perturbing potential as a function of electric field $E$, and does not replace it with a carrier-envelope description. Although apparently the simplest strategy, it is generally impractical as it imposes constraints on the field and other model parameters that are too restrictive to be useful.

$$\begin{aligned}
\imath \hbar \frac{d c_n}{dt}
&=&
-
\sum_i
  c_i
  d_{ni} E 
  \exp \left[ -\imath \left( W_i - W_n \right) t \right]
\\
\textrm{... assume } c_n, d_{ni} && 
\textrm{vary only slowly, so I can integrate just the exponentials }
\\
c_i
&=&
-
 \frac{1}{ \imath \hbar}
 \sum_j
  c_j
  d_{ij} E 
    \frac{
          \exp \left[ -\imath \left( W_j - W_i \right) t \right]
         }
         {-\imath \left( W_j-W_i \right) }
\\
&=&
\frac{1}{ \hbar}
\sum_j
  c_j
  d_{ij} E
    \frac{
          \exp \left[ -\imath \left( W_j - W_i \right) t \right]
         }
         {W_i-W_j}\end{aligned}$$

Now substitute the $c_i$ solution into the $d c_n / dt$ equations, and introduce the shorthand notation $W_{ij} = 