Denote the two ground states and the excited state of the electron  and , respectively. The atom also has overall momentum, so the overall state of the atom is a product state of its internal state and its momentum, as shown in the figure. In the presence of counter-propagating beams of opposite polarization, the internal states experience the atom-light interaction Hamiltonian
where  is the Rabi frequency, approximated to be the same for both transitions. Using the definition of the translation operator in momentum space,
the effect of  on the state  is
This suggests the dressed state  that couples to  is a more convenient basis state of the two ground states. The orthogonal basis state  defined below does not couple to  at all.
The action of  on these states is
Thus,  and  undergo Sisyphus-like cooling, identifying the former as the bright state.  is optically inaccessible and constitutes the dark state. However,  and  are not eigenstates of the momentum operator, and thus motionally couple to one another via the kinetic energy term of the unperturbed Hamiltonian:
As a result of this coupling, the dark state evolves into the bright state with frequency proportional to the momentum, effectively selecting hotter atoms to re-enter the Sisyphus cooling cycle. This nonadiabatic coupling occurs predominantly at the potential minima of the light-shifted coupling state. Over time, atoms cool until they lack the momentum to traverse the sinusoidal light shift of the bright state and instead populate the dark state.