We study a particle on a ring in the presence of various dissipative environments. We develop and solve a variational scheme assuming low frequency dominance. We analyze our solution within a renormalization group (RG) scheme to all orders which reproduces a two loop RG for the Caldeira-Legget environment. In the latter case the Aharonov-Bohm (AB) oscillation amplitude is exponential in $\ensuremath{-}{R}^{2}$, where
R is the ring's radius. For either a charge or an electric dipole coupled to a dirty metal we find that the metal induces dissipation, however the AB amplitude is $\ensuremath{\sim}{R}^{\ensuremath{-}2}$ for large
R, as for free particles. Cold atoms with a large electric dipole may show a crossover between these two behaviors.
A collection of
N Rydberg atoms and a resonant millimeter-wave cavity are shown to exchange energy back and forth at a rate $2(\frac{d}{\ensuremath{\hbar}}){\mathcal{E}}_{0}\sqrt{N}$, where
d is the electric dipole matrix element of the atomic transition and
E0 the "field per photon" in the cavity. This experiment is a demonstration of self-induced Rabi oscillations in a two-level atom system coupled to a single electromagnetic field mode and can also be considered as a very simple illustration of "ringings" in superradiant emission.