I... think I've figured out what they mean.

Absolute Zero is normally defined by every atom in your sample being in its ground state. That's pretty much impossible to achieve, so in every real measurement, you'll have a few atoms in higher energy states. The more high-energy-state atoms you have, the higher the temperature. By that definition, it's impossible to go below Absolute Zero.

But they're not using that definition. What they've done is push more than half of the atoms into higher energy states - no, into the highest possible energy state - and claimed that's a negative temperature. They're using a definition whereby... well, this is the paragraph that explained it to me:

That a system at negative temperature is hotter than any system at positive temperature is paradoxical if absolute temperature is interpreted as an average kinetic energy of the system. The paradox is resolved by understanding temperature through its more rigorous definition as the tradeoff between energy and entropy, with the reciprocal of the temperature, thermodynamic beta, as the more fundamental quantity. Systems with a positive temperature will increase in entropy as one adds energy to the system. Systems with a negative temperature will decrease in entropy as one adds energy to the system.

Right, I think I've actually got it now:

In our normal view of atoms, the energy levels start from zero - the ground state - and head upwards into infinity. There are infinitely many energy states. But, it is possible to induce a situation with finite energy states.

Normal temperature is a measure of disorder. The 'most ordered' state - the state with the lowest entropy - is the one where everything is in the lowest energy state (ie, all identical). Adding more energy moves some of the atoms to higher energy states; a less ordered situation.

But this system has an upper limit as well. Once you get more than half of your atoms into that highest energy state, adding energy can only put more into that state - increasing the ordered nature of the system. You're into a fundamentally different kind of 'temperature', one where adding energy takes you closer to a perfectly ordered system, rather than away from it. Bam - negative temperature.

The key difference is that '-0K' - the perfectly ordered negative state - is not synonymous with '0K'. All your atoms are in the exact opposite state - the top limit, not the bottom.

But still, pretty cool - or hot!

hS