we can assume the following ansatz [@Utami2007] $$W\left[ {k,q;t} \right] =  e^{ - \nu ( t  )} \exp \left(   - i [ {k\bar x (t) + q\bar p (t)}  ] - \frac{1}{2} (k^2+q^2) \sigma _s (t) \right),$$ from which the moment generating function can be computed as $\Lambda[F( t );t_f] = e^{-\nu ( t_f )} $. Substituting this ansatz into the master equation in Eq. , we then need to solve a set of ordinary differential equations for the coefficient functions

\[SEq:cohODE\] $$\begin{aligned}
&\dot \nu_\mathrm{th}  =    i\lambda F ( t ) \left( {\sigma _s- \frac{1}{2}} \right) ,\\
&  {\dot \sigma }_s  = \gamma \left( {\bar n}_\mathrm{th}+ \frac{1}{2}  \right) - \gamma {\sigma _s}   - iF ( t ) \lambda \sigma _s^2 + \frac{ i\lambda F \left( t\right)  }{4} , \\
&\dot \nu_\mathrm{dr}  =     \frac{{i\lambda }}{2} F(t)  ({\bar x}^2 +{\bar p}^2)  ,\\
& \dot {\bar x} =  - \delta \bar p + \sqrt 2 \, \mathrm{Im}f  - iF ( t ) \lambda {\sigma _s}  \bar x- \frac{\gamma }{2}\bar x , \label{SEq:mfGSantcoef-xb}  \\
&  \dot {\bar p}  = \delta \bar x + \sqrt 2 \, \mathrm{Re}f  - i F (t) \lambda {\sigma _s} {\bar p} - \frac{\gamma }{2}\bar p , \label{SEq:mfGSantcoef-pb} 
  \end{aligned}$$

where 