Non-exponential decay of a collective excitation in an atomic ensemble coupled to a one-dimensional waveguide

We study the dynamics of a single excitation coherently shared amongst an ensemble of atoms and coupled to a one-dimensional wave guide. The coupling between the matter and the light field gives rise to collective phenomena such as superradiant states with an enhanced initial decay rate, but also to the coherent exchange of the excitation between the atoms. We find that the competition between the two phenomena provides a characteristic dynamics for the decay of the excitations, and remarkably exhibits an algebraic behavior, instead of the expected standard exponential one, for a large number of atoms. The analysis is first performed for a chiral waveguide, where the problem can be solved analytically, and then is extended to the bidirectional waveguide.