  We address a fundamental issue in quantum mechanics and quantum information
theory, the generation of an entangled pair of qubits that interact solely
through a third, semiclassical degree of freedom, in the framework of cavity
quantum electrodynamics. We show that finite, though not maximal, entanglement
is obtainable in the classical limit, at the price of a diverging effective
interaction time. The optimal atomic entanglement derives from a trade-off
between the atomic entanglement in a sub-wave packet and the purity of the
atomic state. Decoherence by photon loss sets an upper limit on the degree of
excitation of the cavity mode, beyond which the achievable entanglement
decreases as the inverse mean photon number to the sixth power.
