Affine Bloch equation ($\dot{s}=As+c$)

A time-independent Lindblad equation becomes an affine linear equation for the Generalized Bloch representation. If $\mathcal{L}(I)={\sum }_d[V_d,V_d^{†}]$, then $A_{mn}=\mathrm{Tr}({\sigma }_m\mathcal{L}({\sigma }_n))$ and $c_m=\mathrm{Tr}({\sigma }_m\mathcal{L}(I))/N$ for traceless basis indices. The offset vanishes exactly when the dynamics are unital. Equilibria solve $A{s}_{\ast }=-c$, and deviations evolve as $s(t)-s_{\ast }=e^{At}(s(0)-s_{\ast })$.