These are very interesting thoughts! First of all, it is probably necessary to make a distinction between two kinds of interpretations: MWI/decoherence interpretations and collapse interpretations. The reason is that in MWI interpretations, entanglement grows, and irreversible entanglement is at the origin of the lack of interference - while in collapse interpretations, the collapse kills entanglement, and is - as such - responsible for the lack of interference.
The point is that "entanglement" is not, as such, an observable phenomenon; quantum interference is.
As I said already in a few posts, if we have two states of a system, |a> and |b>, then an observable that looks for interference between both is an observable that has as eigenstates |a>+|b> (eigenvalue 1) and |a>-|b> (eigenvalue 0).
In the typical 2-slit experiment, the states |a> and |b> correspond to "through slit 1" and "through slit 2" respectively, and the observable is the intensity observed at, say, a peak of the interference pattern.
When the probability for 1 is 100% and the probability for 0 is 0%, then we have an interference pattern ; when it is 0% for 1 and 100% for 0, we have the complementary interference pattern ; and when it is 50% for 1 and 50% for 0, we have NO interference.
Now, I think that what you call "entanglement preserved" is a special kind of interference which shows as correlations between measurements on two systems. Indeed, if we have an entangled system:
|a>|c> + |b>|d>, it is easy to show that the interference between |a> and |b> as such, disappears, but that the interference between |a>|c> and |b>|d> DOES show up.
A typical way of seeing this, is to test for a correlation between (|a>+|b>) and (|c>+d>).
Indeed, |a>|c> + |b>|d> = 1/2{ (|a>+|b>)(|c>+|d>) + (|a> - |b>)(|c>-|d>)}
This is what's typically happening in "quantum erasure" experiments, where we find coincidences between, say |c>+|d> events and an interference pattern on the a/b side. This correlation (between an interference pattern on the c/d side, and an interference pattern on the a/b side) is usually what is seen as proof of "entanglement" between the a/b and the c/d system.
Now, if this a/b - c/d system interacts with a third system, we will end up with:
|a>|c>|e> + |b>|d>|f>, and now, there will be NO correlation anymore between the |c>+|d> observable, and the |a>+|b> observable (in other words, triggering on the |c>+|d> state will not give rise anymore to the |a>+|b> interference pattern. This is simply because the |e> and the |f> are not an identical state anymore, which can factor out (as was the state of the e/f system before it interacted with the a/b and c/d system).
In the MWI/decoherence view, we simply say the above: the ab-cd system got entangled with another system (the ef system) and as such, destroyed the interference (or coherence) between the ab and cd system. So in this view it is the EXTRA ENTANGLEMENT which destroyed the coherence between the ab and cd system (which served as a proof of their entanglement between them). This is the same mechanism that was responsible for the disappearence of the IP for the ab system alone. When the ab system got entangled with the cd system, the interference pattern of the |a>+|b> observable disappeared, only to be seen as a CORRELATION between the IP of the ab and the cd system. In the same way, the entanglement between the (ab-cd) system and the ef system made this interference disappear, and appear as a correlation between an IP of the ab,cd and ef systems. But as we're not going to do an interference measurement on the ef system, this is just saying that the evidence for entanglement between ab and cd has been destroyed (decohered).
In the projection view, one can consider that the interaction with the e/f system considered a measurement in the {|a>|c> ; |b>|d>} basis, and so we now have, with 50% probability, the PRODUCT state |a>|c> OR the product state |b>|d> (and hence entanglement is GONE).
Of course this view gets in troubles if we are going to test for an interference between the ef system and the ab/cd system.
Now, to answer your questions:
the two photons are given by the |a>|c>+|b>|d> state.
If the cd photon interacts with the atom, we will have:
|a>|e> + |b>|f>, where the e/f states are atom states (the cd photon has disappeared).
So yes, the entanglement is now transmitted to the photon/atom system.
And now, how are we going to emit a photon from these two different atom states, so that we get back our original pair of entangled photons ? The answer is simply: the FINAL STATE of the atom, in the two branches, must be identical. The |e> state must go to a |g>|k> state, and the |f> state must go to a |g>|l> state. So the atom must return to the same state |g> (and the emitted photons to |k> and |l>). If not, the atom will STAY entangled with the two terms, and as such play the role of the decoherence inducing "environment". Because if the final state is the same (for the atom), we will now have:
|a>|g>|k> + |b>|g>|l>
|g> factors out:
|g> (|a>|k> +|b>|l>)
and we now have an entangled photon pair given by a/b and k/l, with the atom "outside" of the entanglement.
It is clear that there is no "projection" version of this thing happening...
cheers,
patrick.
