Effective Hamiltonian in Lindblad Master Equation

If we have a Lindblad master equation of the form \begin{equation} \partial_t \rho = -i [H,\rho] + \sum_n \kappa_n (A_n \rho A_n^\dagger - \frac12 \{A_n^\dagger A_n, \rho\}), \end{equation} where the $A_n$ are operators and the $\kappa_n$ are positive constants, then we can rewrite this as \begin{equation} \partial_t \rho = -i (H_{\mathrm{eff}} \rho - \rho H_{\mathrm{eff}}^\dagger) + \sum_n \kappa_n A_n \rho A_n^\dagger, \end{equation} where the effective non-Hermitian Hamiltonian is given by $H_{\mathrm{eff}} = H - \frac{i}{2} \sum_n \kappa_n A_n^\dagger A_n$. This can be checked by direct substitution. Now writing the density matrix in diagonal form (which is always possible since it is Hermitian) as $\rho = \sum_{j} P_{j} |\psi_j\rangle\langle \psi_j|$, it is easy to check that the first term in the rewritten master equation is obtained by Schrödinger evolution of the wave functions $|\psi_j\rangle$ with the effective Hamiltonian, $\partial_t |\psi\rangle = -i H_{\mathrm{eff}} |\psi\rangle$. The Lindblad master equation can thus be thought of as a coherent evolution given by the Schrödinger equation with a non-Hermitian Hamiltonian, plus incoherent "quantum jumps" given by the "jump" or "refilling" term $\sum_n \kappa_n A_n \rho A_n^\dagger$. Note that since the effective Hamiltonian is not Hermitian, it leads to decay of the wave functions $|\psi_j\rangle$, and the jump term makes this "lost" probability reappear in the state reached after the decay (which is why it is also called a "refilling" term). In many situations, the refilling term can be ignored, for example because it does not contribute to the dynamics within the excited states, but "just" ensures that the lost population is re-injected into the ground state.