p_n^{\uparrow}n(t) = p_{nr}(0) + (\tau_n / \gamma_n)  \cdot  q^{\uparrow}q_n(t)  - \gamma_n^{-1} \cdot Q^{\uparrow}Q_n (t) + \gamma_n^{-1} \cdot  
\left[ \sum_m f_{nm}  \ Q_m^{\downarrow}(t)   + \sum_m g_{nm}  \ Q_m^{\uparrow}(t) \right]

p_n^{\downarrow}(t) = p_{nr}(0) + (\tau_n / \gamma_n)  \cdot  q^{\downparrow}_n(t)  - \gamma_n^{-1} \cdot Q^{\downarrow}_n (t) + \gamma_n^{-1} \cdot  
\left[ \sum_m f_{nm}  \ Q_m^{\uparrow}(t)   + \sum_m g_{nm}  \ Q_m^{\downarrow}(t) \right]