($\omega_i$) of interest, which are usually at integer spacings of the transition frequency ($\omega_A$) from the main pump laser frequency ($\omega_0$). Each of these field components is described usin a standard envelope theory (i.e. as $A_i(t)$) with a time-history, allowing us to simulate pulses as they travel through the Raman medium. The Raman medium is modelled as a two-level atom using a (its) Bloch vector $(u,v,w)$, and this Bloch vector is driven by each combination of spectrally adjacent field components ($\sim \sum A_i A_{-1}$). Each of the field components $A_i$ is driven by the atomic polarization ($\sim v$) in combination with its pair of adjacent field components (i.e. $A_{i-1},
A_{i+1}$).

Each field component evolves as   $$\begin{aligned}
    \partial_z A_j
&=&
\frac{\sigma \omega_j \alpha_{12}}
     { \epsilon_0 c_0}
\left\{
  -
  \left[
     v' 
   -
     \imath u'
  \right]
   A_{j+1}
      \exp\left( +\imath (k'_{j+1} -k'_j) z - \imath \Delta t \right)
 +
  \left[
     v' 
   +
     \imath u'
  \right]
    A_{j-1}
      \exp\left( +\imath (k'_{j-1} -k'_j) z + \imath \Delta t \right)
\right\}
\nonumber
\\
&& ~~~~ ~~~~ ~~~~ ~~~~ 
 - 
  \left( k_j - k_0 \right) A_j\end{aligned}$$

The transition evolves as ($\rho_{12}' \equiv (u',v') $) $$\begin{aligned}
  \partial_t \rho_{12}' 
&=&
  \left(
     -\gamma_2 + \imath \Delta + 2 \imath g' \sum_j A_j^* 