Lasers do this. The essential idea is that an incoming photon (of some specified energy E) knocks a bit of energy (specifically also E) out of an excited particle which leaves as another photon. To make this happen, you need more particles in an excited state than in the ground state[1] which is exactly the same condition necessary for negative temperature.
Take an atom that somehow only has two energy levels, 0 and 1. Connect the atom to a heat sink at temperature T. At very low T that atom almost always has energy 0. We know the energy of each atom very well, they're (almost) all 0. People say: there's very little entropy. At very high T the atom has expected energy nearly 0.5, and the probabilities of energy 0 or energy 1 are nearly equal. So that's maximum entropy. We're as ignorant as it's possible to be.
At negative temperature the expected energy of each atom is >0.5. But as the expected energy approaches 1.0, we know the energy of each atom very well. They're (almost) all 1. That's weird. It's weird enough that you can't assign a positive temperature to these atoms.
Physically, you can get to negative temperature by sneaking atoms into the 1 state. Pumping a laser is an example. But you can't get to negative temperature by just heating with finite-temperature heaters.
Entropy can be measured in bits. If we have 10 two-level atoms at extremely high temperature, that's 10 bits of entropy. The state might be 10 1100 1110 or 01 1101 0111 or any of 2^10 possibilities. On the other hand if we have 10 two-level atoms at extremely low positive temperature, the state is usually 00 0000 0000 and the entropy is close to 0 bits.
"Entropy" in bits is just the size of the random number you need in order to represent the system.
[1]: https://en.wikipedia.org/wiki/Population_inversion