Thanks for the great example. It seems more than ever to me as though "entangled" isn't quite the right word for the phenomenon. There is no co-dependence. The two atoms have just been configured/seeded to perform with some predictable similarity. I.e., only their states are correlated.DrChinese has given the example of full entanglement swapping, but there's a kind of 'half-way house' that might help to shed some light.
Imagine a perfect optical cavity (the experiments were done in very high-Q microwave cavities). Now take a 2 level atom in its excited state (Rydberg atoms can be used as a reasonable approximation to a 2 level atom). Fire this atom through the cavity with a specific transit time such that the field and atom become perfectly entangled.
Now suppose we live in an ideal world and we can maintain the entanglement between the cavity field and the atom. Go make a cup of tea. Ship the atom off to the outer moons of Saturn.
Now take a second atom prepared in its ground state and fire this through the cavity with a different tailored transit time through the cavity. Tailor this time just right and after this second atom has gone through the cavity the 2 atoms are now entangled and the cavity field 'decoupled'.
The two atoms have never directly interacted - and (if we can maintain the entanglement long enough) the two atoms can be fired through the cavity years apart.
OK that's wildly fanciful in terms of shipping things off to Saturn and maintaining entanglement for years - but the experiments have been performed (although with more modest parameters).
Question: Does it have to be the same "perfect optical cavity," or could these two atoms be "entangled" via two identical cavities in distant locations?