  We investigate the quantum optical properties of Maxwell's two-dimensional
fish eye lens at the single-photon and single-atom level. We show that such a
system mediates effectively infinite-range dipole-dipole interactions between
atomic qubits, which can be used to entangle multiple pairs of distant qubits.
We find that the rate of the photon exchange between two atoms, which are
detuned from the cavity resonances, is well described by a model, where the
photon is focused to a diffraction-limited area during absorption. We consider
the effect of losses on the system and study the fidelity of the entangling
operation via dipole-dipole interaction. We derive our results analytically
using perturbation theory and the Born-Markov approximation and then confirm
their validity by numerical simulations. We also discuss how the
two-dimensional Maxwell's fish eye lens could be realized experimentally using
transformational plasmon optics.
