We present a quantum treatment of Sisyphus cooling in two dimensions.
The steady state is given by the populations of the energy eigenstates
for the atomic motion in the potential created by the atom-light inte
raction. In addition to characterizing the momentum and position distr
ibutions, we show that resonant tunnelling between adjacent optical po
tential wells leads to a sizable variation of the lowest-energy state
population, as a function of the field parameters.