Subradiant emission from regular atomic arrays: universal scaling of decay rates from the generalized Bloch theorem

The Hermitian part of the dipole-dipole interaction in infinite periodic arrays of two-level atoms yields an energy band of singly excited states. In this Letter, we show that a dispersion relation, $\omega_k-\omega_{k_\ex} \propto (k-k_{\ex})^s$, near the band edge of the infinite system leads to the existence of subradiant states of finite one-dimensional arrays of $N$ atoms with decay rates scaling as $N^{-(s+1)}$. This explains the recently discovered $N^{-3}$ scaling and it leads to the prediction of power law scaling with higher power for special values of the lattice period. For the quantum optical implementation of the Su-Schrieffer-Heeger (SSH) topological model in a dimerized emitter array, the band-gap-closing inherent to topological transitions changes the value of $s$ in the dispersion relation and alters the decay rates of the subradiant states by many orders of magnitude.