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\nthree-dimensional magneto-optical trap (MOT), by numerically solving the\noptical Bloch equations. We focus on the case where there are dark states\nbecause the atom is driven on a "type-II" system where the angular momentum of\nthe excited state, $F'$, is less than or equal to that of the ground state,\n$F$. For these systems we find that the force in a three-dimensional light\nfield has very different behaviour to its one dimensional counterpart. This\ndiffers from the more commonly used "type-I" systems ($F'=F+1$) where the 1D\nand 3D behaviours are similar. Unlike type-I systems where, for red-detuned\nlight, both Doppler and sub-Doppler forces damp the atomic motion towards zero\nvelocity, in type-II systems in 3D, the Doppler force and polarization gradient\nforce have opposite signs. As a result, the atom is driven towards a non-zero\nequilibrium velocity, $v_{0}$, where the two forces cancel. We find that\n$v_{0}^{2}$ scales linearly with the intensity of the light and is fairly\ninsensitive to the detuning from resonance. We also discover a new\nmagneto-optical force that alters the normal MOT force at low magnetic fields\nand whose influence is greatest in the type-II systems. We discuss the\nimplications of these findings for the laser cooling and magneto-optical\ntrapping of molecules where type-II transitions are unavoidable in realising\nclosed optical cycling transitions.\n