What does "generator" mean in the master equation?

Question

I seem to read a lot of times that some materials called this $\mathcal{L}$ in the equation(Lindblad master equation) below as the generator: $$ \mathcal{L} \rho=-i[H, \rho]+\sum_{\alpha}\left(V_{\alpha} \rho V_{\alpha}^{\dagger}-\frac{1}{2}\left\{V_{\alpha}^{\dagger} V_{\alpha}, \rho\right\}\right). $$ I only know the generator in group theory means the minimum component that can span the whole group by the multiplication action, but I can't see what does the generator there means?

Answer 1

The idea is that $$ \frac{d}{dt}\rho=\mathcal{L}\rho\qquad \Leftrightarrow\qquad \rho(t)=\exp(t\mathcal{L})\rho(0). $$ In that sense, the Lindbladian $\mathcal{L}$ generates evolution through $$\rho(dt)\approx \rho(0)+dt \mathcal{L}\rho(0),$$ which is the same sense as an infinitesmal generator of a Lie group. You might be familiar with something like a rotation of a vector $\mathbf{u}(0)$ by some rotation angle $|\mathbf{r}|$ about some axis pointing in the direction $\mathbf{r}$: $$R(\mathbf{r})\mathbf{u}(s)=\exp(i \mathbf{J}\cdot\mathbf{r})\mathbf{u} (0)\qquad\Leftrightarrow\qquad \mathbf{u}(d\mathbf{r})\approx \mathbf{u}(d\mathbf{r})+i \left(\mathbf{J}\cdot d\mathbf{r}\right) \,\mathbf{u}(0);$$ then, the operators $\mathbf{J}=(J_1,J_2,J_3)$ are the generators of rotations.

The idea is explicitly explained in Lindblad's original paper, from which the name ``generators of quantum dynamical semigroups'' derives, where the relevant fact is that $\lim_{t\to 0}\rho(t)=\rho(0)$.