Bichromatic magneto-optical trapping for $ J\rightarrow J,J-1 $ configurations

A magneto-optical trap (MOT) of atoms or molecules is studied when two lasers of different detunings and polarization are used. Especially for $J\rightarrow J,J-1$ transitions, a scheme using more than one frequency per transition and different polarization is required to create a significant force. Calculations have been performed with the simplest forms of the $J\rightarrow J-1$ case (i.e. $J''=1 \rightarrow J'=0$) and $J\rightarrow J$ case (i.e. $J''=1/2 \rightarrow J'=1/2$). A one dimensional (1D) model is presented and a complete 3D simulation using rate equations confirm the results. Even in the absence of Zeeman effect in the excited state, where no force is expected in the single laser field configuration, we show that efficient cooling and trapping forces are restored in our configuration. We study this mechanism for the C$_2^-$ molecular anion as a typical example of the interplay between the two simple transitions $J \rightarrow J,J-1$.