collective spin basis are degenerate, with degeneracy given by: $\alpha_J = 1,\dots,  { {N \choose J} }-{ {N \choose J-1} }$. It is easy to observe in this basis that the states ${|J,-J,\alpha_J\rangle}$ are decoherence free, and therefore span the *decoherence-free subspace* (DFS).

The DFS has a dimension of ${ {N \choose N/2} }$ (assuming even atom number $N$), and is composed of all the possible states which are antisymmetric with the permutation of two atoms. Thus, an alternative way of characterizing the DFS is to consider all possible (tensor products) of singlet states $${|A_{m,n}\rangle} = \left({|e\rangle}_m \otimes {|g\rangle}_n-{|g\rangle}_m \otimes {|e\rangle}_n\right)/\sqrt{2}\,,$$ where $m,n$ denote the atomic positions of the pair of atoms that form the singlet. This characterization makes it more difficult to describe an orthonormal basis of the DFS. However, we show in the next section that it is convenient to define our computational subspace.

Quantum Zeno dynamics using Liouvillian perturbation theory {#sec2}
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We are interested in the regime where the collective dissipation induced by $\mathcal{L}_\mathrm{D}$, with characteristic timescale $\Gamma_{\oned}^{-1}$, dominates over any possible perturbation of the system, $\mathcal{L}_\mathrm{pert}$, with characteristic timescale $\tau\gg\Gamma_{\mathrm{1d}}^{-1}$. Under these assumptions, any state outside of the DFS will only be virtually populated due 