to the strong dissipation and therefore the dynamics will be restricted to the slow subspace, i.e., the DFS. Mathematically, we formalize this intuitive picture by defining a projection superoperator $\mathbb{P}$ (with $\mathbb{P}^2 = \mathbb{P}$) satisfying: $\mathbb{P}{\cal L}_\DD={\cal L}_\DD \mathbb{P}=0$ that projects out the fast dynamics yielding only the effective evolution in the slow subspace. It is then possible to integrate out the fast dynamics (see Appendix A) arriving to an effective master equation given by $$\frac{\partial \PP\rho}{\partial t}=\mathcal{L}_\mathrm{eff} [\PP\rho]= \Big(\PP  \mathcal{L}_\mathrm{pert} \PP +\PP  \mathcal{L}_\mathrm{pert} \QQ \frac{1}{{-\cal L}_\DD}\QQ  \mathcal{L}_\mathrm{pert} \PP +\mathrm{O}\left(\tau^{-3}/\Gamma_\oned^2 \right)\Big) \rho\,.\label{eq:effemastereq}$$ This result to first order (left hand term in the brackets) accounts for the ideal *Quantum Zeno dynamics* [@zanardi97a; @lidar98a; @facchi02a]. The second order in perturbation theory then yields correction terms mainly coming from slightly populating the (super)radiant states.

In our case, there will be two types of perturbations, namely,

-   The Hamiltonian $\mathcal{L}_\mathrm{pert}[\cdot]= - \rmi \left[H_{\mathrm{las}}, \cdot \right]$ to control the atomic state. This results to first order in an effective Hamiltonian $H_{\mathrm{eff}}=\mathcal{P} H_{\mathrm{las}} \mathcal{P}$ that couples only atomic states within the DFS. Here, we introduced the projection onto the DFS for pure states $\mathcal{P} = \sum_i {|\mathrm{d}_i\rangle}{\langle \mathrm{d}_i|}$, where the states ${|\mathrm{d}_i\rangle}$ form an orthonormal basis 