  With subrecoil-laser-cooled atoms one may reach nano-Kelvin temperatures
while the ergodic properties of these systems do not follow usual statistical
laws. Instead, due to an ingenious trapping mechanism in momentum space,
power-law-distributed sojourn times are found for the cooled particles. Here,
we show how this gives rise to a statistical-mechanical framework based on
infinite ergodic theory, which replaces ordinary ergodic statistical physics of
a thermal gas of atoms. In particular, the energy of the system exhibits a
sharp discontinuous transition in its ergodic properties. Physically this is
controlled by the fluorescence rate, but more profoundly it is a manifestation
of a transition for any observable, from being an integrable to becoming a
non-integrable observable, with respect to the infinite (non-normalised)
invariant density.
