Quasi-flat band enables subradiant two-photon bound states

We study theoretically the radiative lifetime of bound two-particle excitations in a waveguide with an array of two-level atoms, realising a 1D Dicke-like model. Recently, Zhang et al. [arXiv:1908.01818] have numerically found an unexpected sharp maximum of the bound pair lifetime when the array period $d$ is equal to $1/12$th of the light wavelength $\lambda_0$]. We uncover a rigorous transformation from the non-Hermitian Hamiltonian with the long-ranged radiative coupling to the nearest-neigbor coupling model with the radiative losses only at the edges. This naturally explains the puzzle of long lifetime: the effective mass of the bound photon pair also diverges for $d=\lambda_0/12$, hampering an escape of photons through the edges. We also link the oscillations of the lifetime with the number of atoms to the nonmonotous quasi-flat-band dispersion of the bound pair.