  We introduce a complete Bell measurement on atomic qubits based on two photon
interactions with optical cavities and discrimination of coherent states of
light. The dynamical system is described by the Dicke model for two three-level
atoms interacting in two-photon resonance with a single-mode of the radiation
field, which is known to effectively generate a non-linear two-photon
interaction between the field and two states of each atom. For initial coherent
states with large mean photon number, the field state is well represented by
two coherent states at half revival time. For certain product states of the
atoms, we prove the coherent generation of GHZ states with two atomic qubits
and two orthogonal Schr\"odinger cat states as a third qubit. For arbitrary
atomic states, we show that discriminating the two states of the field
corresponds to different operations in the Bell basis of the atoms. By
repeating this process with a second cavity with a dephased coherent state, we
demonstrate the implementation of a complete Bell measurement. Experimental
feasibility of our protocols is discussed for cavity-QED, circuit-QED and
trapped ions setups.

Quantum Physics