Three-dimensional Doppler, polarization-gradient, and magneto-optical forces for atoms and molecules with dark states

Abstract

We theoretically investigate the damping and trapping forces in a three-dimensional magneto-optical trap (MOT), by numerically solving the optical Bloch equations. We focus on the case where there are dark states because the atom is driven on a ‘type-II’ system where the angular momentum of the excited state, , is less than or equal to that of the ground state, F. For these systems we find that the force in a three-dimensional light field has very different behaviour to its one dimensional counterpart. This differs from the more commonly used ‘type-I’ systems () where the 1D and 3D behaviours are similar. Unlike type-I systems where, for red-detuned light, both Doppler and sub-Doppler forces damp the atomic motion towards zero velocity, in type-II systems in 3D, the Doppler force and polarization gradient force have opposite signs. As a result, the atom is driven towards a non-zero equilibrium velocity, v0, where the two forces cancel. We find that scales linearly with the intensity of the light and is fairly insensitive to the detuning from resonance. We also discover a new magneto-optical force that alters the normal MOT force at low magnetic fields and whose influence is greatest in the type-II systems. We discuss the implications of these findings for the laser cooling and magneto-optical trapping of molecules where type-II transitions are unavoidable in realising closed optical cycling transitions.