  We present a probabilistic scheme for generating and purifying
maximally-entangled states of two atoms inside an optical cavity via no-photon
detection in the output cavity mode, where ideal detectors may not be required.
The intermediate mixed states can be continuously "filtered" so as to violate
Bell inequalities in a parametrized manner. The scheme relies on an additional
strong-driving field that yields unusual dynamics in cavity QED experiments,
simultaneously realizing Jaynes-Cummings and anti-Jaynes-Cummings interactions.
