This is an interesting discussion, which I unfortunately don't know enough
solid state physics to contribute to.
However my instincts are that some of these ideas are based on a macroscopic
model which won't work at the level we are talking about. These atomic
vibrations are themselves heat. The notion of going in and producing
counter vibrations which will cancel these smacks of violating the Third
Law.
Of course refrigerators do exist, and there is nothing wrong with heat
flowing from a cold to a hot object as long as entropy somehow balances
out. But when you are talking about heat flowing from a cold to a hot
molecule, it is hard to see how to do it.
I know that one way refrigerators work is by evaporation, which in effect
means that we filter molecules, allowing the high speed ones to go away
and keeping just the low speed ones. We don't actually slow any down in
this process but we do end up with low energy molecules. However it is
not clear how to apply this to the atom-in-a-box example.
Hal