  We study theoretically and experimentally the competing blockade and
anti-blockade effects induced by spontaneously generated contaminant Rydberg
atoms in driven Rydberg systems. These contaminant atoms provide a source of
strong dipole-dipole interactions and play a crucial role in the system's
behavior. We study this problem theoretically using two different approaches.
The first is a cumulant expansion approximation, in which we ignore third-order
and higher connected correlations. Using this approach for the case of resonant
drive, a many-body blockade radius picture arises, and we find qualitative
agreement with previous experimental results. We further predict that as the
atomic density is increased, the Rydberg population's dependence on Rabi
frequency will transition from quadratic to linear dependence at lower Rabi
frequencies. We study this behavior experimentally by observing this crossover
at two different atomic densities. We confirm that the larger density system
has a smaller crossover Rabi frequency than the smaller density system. The
second theoretical approach is a set of phenomenological inhomogeneous rate
equations. We compare the results of our rate equation model to the
experimental observations in [E. A. Goldschmidt, et al., PRL 116, 113001
(2016)] and find that these rate equations provide quantitatively good scaling
behavior of the steady-state Rydberg population for both resonant and
off-resonant drive.
