1 
                - \left( 2 \frac{W_{21}}{ W_{i1} } 
                         + \frac{W_{21}^2}{ W_{i1}^2}
                  \right)
                   \frac{W_{i1}^2}{ \left( W_{i1}^2 -\omega_0^2 \right)}
                + ...
           \right]
  \right\}
\\
&\approx&
      \frac{1}{ \hbar}
  \sum_i
  \left\{
      \left| d_{1i} \right|^2
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
      \left| d_{2i} \right|^2
    \frac{ W_{i1} + W_{21} }
         {W_{i1}^2 -\omega_0^2 }
           \left[ 
                  1 
                - 2 \frac{W_{21}}{ W_{i1} }
                    \frac{W_{i1}^2 }{ \left( W_{i1}^2 -\omega_0^2 \right)}
           \right]
  \right\}
\\
&\approx&
      \frac{1}{ \hbar}
 \sum_i
 \left\{
  \left[
      \left| d_{1i} \right|^2
 - 
      \left| d_{2i} \right|^2
 - 
      \left| d_{2i} \right|^2
      \frac{W_{21}}{ W_{i1}}
  \right]
    \frac{ W_{i1} }
         {W_{i1}^2-\omega_0^2}
 - 
  \left[
      2 \left| d_{2i} \right|^2
      \frac{W_{21}}{ W_{i1}}
  \right]
      \frac{ W_{i1}^3 }
           { \left( W_{i1}^2 -\omega_0^2 \right)^2 }
 \right\}
\label{eqn-alpha-stark-approx}\end{aligned}$$

Ignoring the $W_{21}$ transition frequency terms, this is just the difference in the energy shifts of the two levels with field intensity. This is a purely [*real*]{} quantity, with no imaginary part.

Two photon Bloch equations
--------------------------

From now on, I use $\mathscr{I}$ (a script “I”) to represent $E^2$ if following on from \[sss-single-Efield\] and eqns.(\[eqn-single-E-DcnDt\], \[eqn-single-E-alpha\]); or to represent $2\left| A \right|^2$ if following on from \[sss-single-Eenvelope\] and eqns.(\[eqn-single-A-DcnDt\], \[eqn-single-A-alpha\]). Since it is the envelope-carrier description of the field $E$ which is most useful (in \[sss-single-Eenvelope\]), for most purposes $\mathscr{I} = 2\left| A \right|^2$ 