Collective excitations in circular atomic configurations, and single-photon traps

Correlated excitations in a plane circular configuration of identical atoms with parallel dipole moments are investigated. The collective energy eigenstates, their level shifts and decay rates are computed utilizing a decomposition of the atomic state space into carrier spaces for the irreducible representations of the symmetry group $\ZZ_N$ of the circle. It is shown that the index $p$ of these representations can be used as a quantum number analogously to the orbital angular momentum quantum number $l$ in hydrogen-like systems. Just as the hydrogen s-states are the only electronic wave functions which can occupy the central region of the Coulomb potential, the quasi-particle corresponding to a collective excitation of the atoms in the circle can occupy the central atom only for vanishing $\ZZ_N$ quantum number $p$. If a central atom is present, the $p=0$ state splits into two and shows level-crossing at certain radii; in the regions between these radii, damped Rabi oscillations between two "extreme" $p=0$ configurations occur. The physical mechanisms behind super- and subradiance at a given radius and the divergence of the level shifts at small interatomic distances are discussed. It is shown that, beyond a certain critical number of atoms in the circle, the lifetime of the maximally subradiant state increases exponentially with the number of atoms in the configuration, making the system a natural candidate for a {\it single-photon trap}.