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, $F^{\prime} $ , 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 ( $F^{\prime} =F+1$ ) 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, v _0 , where the two forces cancel. We find that ${v}_{0}^{2}$ 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.

Keywords

• laser cooling
• magneto-optical trapping
• cold atoms
• cold molecules