So I read the actual paper and now I want to smack the author of that link silly. He wrote, "For example, while atoms are usually pulled downward by gravity, those below absolute zero float upwards instead."
There is no mention of anything of the sort in the actual paper. This is complete fiction.
So what do they actually mean by negative temperature? The definition they are using for temperature is that the probability of being in a state i with energy E_i is proportional to exp(-E_i / (k_B * T)), where k_B is Boltzmann’s constant. You might note that since we're talking about discrete energy states this is an inherently quantum mechanical definition, but the idea is simply that there is a decaying exponential distribution that favors lower energy states. So what if T goes negative? Well, it's not a decaying exponential anymore, and for that to make any sense you need a system will a well-defined upper bound energy, which the paper spends a while talking about.
But is this "negative temperature" distribution the same as being cold? Actually, no. The paper even says, "...emphasizing that negative temperature states are hotter than positive temperature states, i.e., in thermal contact, heat would flow from a negative to a positive temperature system."
So is this just semantics? Maybe in some sense. It's interesting because the negative temperature state is actually stable (under the right conditions), not some rapidly-decaying pseudo-state. They mention some potential applications in the paper, but this isn't anything that's going to change the world. But I guess that's less exciting than antigravity...
