The medium producing the laser is at a negative temperature.
This is the key to the production of lasers where the atoms in the medium are in an inverted statistical distribution where more atoms are in the excited state than in the ground state.
When I saw the head and first couple posts. This is what I immediately thought of: Population inversion. I don't think the "temperature of the beam" has any real meaning. (The laser beam is a coherent, monochromatic beam of photons. Temperature is not really a good description of it.)
There are several kinds of temperature. We mostly think of a collisional temperature related (in a gas) to the average kinetic energy of the gas particles.
As some of these videos have noted, there is also an "excitation temperature" for quantum mechanical systems. For a two level system (with a lower state "0" and upper state "1") the ratio of the number of atoms in each state is:
Where "n" is the number in each state, "g" is the "quantum degeneracy" (assume both are the same for simplicity), "E" is the (positive) difference in energy between the two states (the excitation energy) and "k" is the Boltzmann constant that translates the units of energy and temperature.
For positive temperatures, as T goes to 0 the exponential goes to zero and all are in the lower state (none in the upper state). As T goes to infinity, E/kT goes to 0 and exp(0) = 1, so the ratio of populations is the ratio of quantum degeneracies. (The ground state of hydrogen is an s(1/2) state, so it has a degeneracy of 2, etc.)
If there *are* more in the upper state than the lower state, then the excitation temperature is negative, starting with negative infinity, since both positive and negative are reached at the limit where the populations are equal, one from "above" and one from below. Negative zero (T=-0) is reached when the population is fully inverted (n_0 = 0) and the ratio goes to infinity and exp (-E/kT) goes to + infinity.
In systems where the atoms are in collisional equilibrium the kinetic and thermal temperatures are the same, and the formula can be used to derive the populations based on the kinetic temperature.
