Ordinary systems have no upper limit on their energy. A gas can have its molecules made faster without end, an oscillator its amplitude made larger without end.
A two-level system, though, does have a limit. Once every particle has been raised to the upper level, no more will go in. The energy is shut inside the range from 0 to N\varepsilon.
The figure is that system. There is a ceiling above. From that fact alone, the meaning of temperature changes partway. This piece is about where the definition of temperature leads if it is simply followed through.
There are C(N, n) states with n of the N particles in the upper level, and their energy is n\varepsilon. The count is a binomial coefficient, nothing more.
The figure is that shape. At E equal to 0, and again at N\varepsilon, there is only one arrangement. The count is largest in the middle.
In an ordinary system the count of states keeps growing with the energy. Here it starts to fall partway along. That is what having a ceiling means.
The definition of temperature was \dfrac{1}{T} = \dfrac{\partial S}{\partial E}. Since S = k\log W, the slope of the curve in the figure is 1/T itself.
On the left half the slope is positive, so T is positive. At the top the slope is 0, making 1/T zero, which is to say T infinite.
On the right half the slope turns negative. Put energy in and the count of states goes down rather than up. So 1/T is negative, and T is negative. It looks like a strange conclusion, and yet not one line of the definition was rewritten.
A negative temperature is not colder than absolute zero. It is hotter than infinity.
The figure draws \beta = \dfrac{1}{kT} against the energy. It comes down from a large positive value at the left, passes 0 at the top, and goes to a large negative value at the right. Nowhere does it jump. What jumps is T, leaping from +\infty to -\infty. Seen through \beta there is no leap at all.
Hot and cold order themselves by \beta: the smaller \beta is, the hotter. Positive temperatures, then infinity, then negative temperatures. Join a system at negative temperature to one at positive temperature and heat flows from the negative one to the positive one.
A spin in a magnetic field is a two-level system. Along the field is the lower level, against it the higher, and in equilibrium more sit in the lower one.
Now reverse the field quickly. Upper and lower exchange meanings. Before the spins have time to turn, the level holding the most has become the higher one. This is called an inverted population, and the right of the figure is that state. Its temperature is negative.
In 1951 Purcell and Pound made exactly this with the nuclear spins of lithium fluoride. A laser gives out light for the same reason: the same inverted population. It is allowed only in systems with a ceiling above, and cannot be made in a gas.