I wonder if this answer to What would happen if we supercool and then superheat an atom very abruptly? goes far enough.
I almost wrote the comment:
I don't think we can even talk about the temperature of a single atom, at least based on its kinetic energy. Now if there were several electrons in one atom simultaneously in higher orbits at the same time, then it might be possible.
but then decided to cast this in the form of a question.
One way to think of the temperature of a group or ensemble of atoms can be the average of a particular energy. There can be more than one temperature; e.g. kinetic energy, the distribution of excited states and nuclear spins (What is the ortho/para issue with LH2 as a fuel?) If they are different we can say that they are out of equilibrium, but we can still talk about each as a different temperature in cases where coupling between different partitions is slow compared to the timescale of interest.
In a strongly interacting quantum system like an atom, there can be some processes that can excited several electrons to higher energy states at once.
Question: Has a concept of temperature ever been (usefully) defined in the context of a single atom?
This would likely be an extreme situation and may not happen in conventional reactions, maybe an intense and ultra-short laser pulse or nuclear decay or reaction, or possibly in the case of some explosive reaction.
update: comments mention a related question in Physics SE Can a single molecule have a temperature? with several answers, though they don't directly answer this question because I'm trying to find out if this "has ever been defined in the context of a single atom", and not asking if you think it should/shouldn't or can/can't be. It is possible that at some point in some, (and possibly extreme) experimental contexts this may have been workable.
The answer there that I do find most helpful is @JohnRennie's. Still though, I'm asking primarily about excitations within a single atom rather than the rotational/vibrational states of a large molecule.