  We present a new paradigm in the field of cavity QED by bringing out
remarkable features associated with the avoided crossing of the dressed state
levels of the Jaynes Cummings model. We demonstrate how the parametric
couplings, realized by a second order nonlinearity in the cavity, can turn the
crossing of dressed states into avoided crossings. We show how one can generate
coherence between the avoided crossing of dressed states. Such coherences
result, for example, in quantum beats in the excitation probability of the
qubit. The quality of quantum beats can be considerably improved by
adiabatically turning on the parametric interaction. We show how these avoided
crossings can be used to generate superpositions of even or odd Fock states
with the remarkable property of equal weights for the states in superposition.
The fidelity of generation is more than 95\%. In addition, we show strong
entanglement between the cavity field and the qubit with the concurrence
parameter exceeding 90\%.
