Magneto-optical trapping forces for atoms and molecules with complex level structures

Laser cooling and magneto-optical trapping of molecules typically involves multiple transitions driven by several laser frequencies. We analyze how magneto-optical trapping forces depend on the angular momenta, $F_l$ and $F_u$, and the g-factors, $g_l$ and $g_u$, of the lower and upper states. When $F_l > F_u$thepolarizationsmustbereversedrelativetocaseswhere$F_u \ge F_l$. The correct choice of circular polarization depends on the sign of $g_u$ but not on the sign of $g_l$. If $g_u$ is zero there is no trapping force, and the trapping force is very weak whenever $g_u$ is small compared to $g_l$, which it usually is when the cooling transition is the ${}^2\mathrm{\Sigma }$ to ${}^2{\mathrm{\Pi }}_{1\mathrm{/}2}$ transition of a molecule. For some molecules, mixing of the excited ${}^2{\mathrm{\Pi }}_{1\mathrm{/}2}$ state with a nearby ${}^2\mathrm{\Sigma }$ excited state can greatly increase $g_u$, leading to stronger trapping forces. A strong trapping force can also be produced by rapidly and synchronously reversing both the magnetic field and the laser polarizations. We simulate a recent experiment on magneto-optical trapping of SrF molecules, and suggest that an alternative choice of laser beam polarizations will strengthen the trapping force.