Radiative cooling force in atoms with multiplet structure

This paper extends the calculation of laser cooling forces on atoms with magnetic degeneracy to the case where the excited state is composed by a multiplet of levels with arbitrary energy separation. The sub-Doppler force that arises in the ${\sigma }_{+},{\sigma }_{-}$ field configuration is found to be strongly affected if other atomic levels lie close to the excited level of the cooling transition. The paper examines in detail the case in which the excited multiplet comes from spin-orbit coupling of angular momentum $L=1\mathrm{}$ and spin $S=1,$ forming the $J_e=0,$ $J_e=1,$ and $J_e=2\mathrm{}$ energy levels. The cooling transition $J_g=1\mathrm{}\leftrightarrow J_e=2\mathrm{}$ shows the sub-Doppler structure due to magnetic degeneracy. The force is modified by the two other transitions (if sufficiently close), even if these do not provide any mechanical effect on the atom when acting alone. The paper presents a detailed scheme for the construction of the relevant optical Bloch equations for the present case, which can be generalized to treat more complex atomic structures. The solutions to these equations are then discussed; several cases of energy separation are worked out and the ensuing graphs of the radiative force are shown. The calculations were carried out in both low- and strong-field regimes.