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

A magneto-optical trap (MOT) of atoms or molecules is studied when two lasers of different detunings and polarization are used. Especially for $J\to J,J-1$ 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 $J\to J-1$ case (i.e., $J^{\prime \prime }=1\to J^{\prime }=0$) and $J\to J$ case (i.e., $J^{\prime \prime }=1/2\to J^{\prime }=1/2$). A one-dimensional (1D) model is presented and a complete 3D simulation using rate equations confirms 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 ${\mathrm{C}}_2^{-}$ molecular anion as a typical example of the interplay between the two simple transitions $J\to J,J-1$.

Figure 1

One-dimensional laser cooling in the $J^{\prime \prime }=1\to J^{\prime }=0$ configuration ($g^{\prime \prime }<0$). Green (light gray) and red (dark gray) respectively refer to lasers propagating to the left ($L^{-}$) and right ($L^{+}$). Dotted lines are used for $M^{\prime \prime }=-1$ levels and the corresponding driving transition ${\sigma }^{+}$ whereas solid lines represent $M^{\prime \prime }=+1$ levels and transitions ${\sigma }^{-}$. The total force $\mathrm{F}={\mathrm{F}}^{+}+{\mathrm{F}}^{-}$, resulting from lasers coming from $z>0$ and $z<0$, is given by Eq. (3) which combines the coupling strengths ${\gamma }_{\pm 1}^{\pm }$ (schematically indicated by the thickness of the vertical arrows) and the amounts of population ${\rho }_{\pm 1}$ in the $M^{\prime \prime }=\pm 1$ levels (schematically indicated by the thickness of lines representing the internal states). Panels (a) and (b): standard case with one ${\sigma }^{+}\mathrm{and}{\sigma }^{-}$ red-detuned laser field. Panels (c) and (d): an extra ${\sigma }^{-}\mathrm{and}{\sigma }^{+}$ laser field on resonance (for $z=v=0$) is added to the standard case. For trapping (in $v=0$), see panels (a) and (c): the energy of $M^{\prime \prime }=\pm 1$ depends on the position because of the Zeeman effect. For cooling (in $z=0$), see panels (b) and (d) where the Doppler effect shifts the two $M^{\prime \prime }=\pm 1$ states in the same way. Because of the two directions of propagation, there are two sets (red and green) of shifted states.

Figure 2

Upper panels: trapping and cooling forces produced by the bichromatic configuration with ${\sigma }^{+}$ and ${\sigma }^{-}$ polarization and $g^{\prime \prime }=-1$. The results correspond to the case of Figs. 1 and 1. Lower panels: for comparison, forces obtained with the standard type-I MOT $J^{\prime \prime }=0\to J^{\prime }=1$ with ${\sigma }^{+}\mathrm{and}{\sigma }^{-}$ single laser beam configuration and $g^{\prime }=-1$. Solid lines: analytical 1D simulation. Squared dots: Monte Carlo 1D simulation. Open circle: Monte Carlo 3D simulation. Simulations are performed with saturation parameters $s=1$ detunings $\delta =-\mathrm{\Gamma }$ and $\delta =0$ when a second laser is present.

Figure 3

Dependence of the trapping (left) and cooling force (right) on different values of the second laser detuning. Parameters are similar to those used in Fig. 2 (upper panel: 3D simulation; lower: 1D analytical solutions).

Figure 4

Trapping and velocity forces with dual laser frequency and polarization on the $J^{\prime \prime }=1/2\to J^{\prime }=1/2$ configuration with $g^{\prime }=0$ and $g^{\prime \prime }=1$. The other parameters are the same as those used in Fig. 2. The 1D case (solid lines) is solved analytically while the 3D case (open circles) results from the Monte Carlo simulation.

Figure 5

Three different bichromatic MOT schemes for ${\mathrm{C}}_2^{-}$ using red-detuned cooling lasers driving the transitions (i) $J^{\prime \prime }=1/2\to J^{\prime }=1/2$, (ii) $J^{\prime \prime }=3/2\to J^{\prime }=1/2$, and (iii) their combination. Symbols $J, N$, and $\pm$ respectively denote the total angular momentum, the rotational quantum number, and the total parity of the molecular wave function. For each scheme, the two lower levels belong to $X^2\mathrm{\Sigma }\left(v^{\prime \prime }\right)$ while the upper level belongs to $B^2\mathrm{\Sigma }(v^{\prime }=0)$.

Figure 6

Acceleration as a function of position and speed (in SI units) for three different bichromatic configurations using ${\mathrm{C}}_2^{-}$ parameters. These configurations correspond to the standard MOT lasers, producing no force alone, and lasers tuned on (i) $J^{\prime \prime }=1/2\to J^{\prime }=1/2$ (left), (ii) $J^{\prime \prime }=3/2\to J^{\prime }=1/2$ (center), and (iii) their combination (right).