Prospects for Doppler cooling of three-electronic-level molecules

Analogous to the extension of laser cooling techniques from two-level to three-level atoms, Doppler cooling of molecules with an intermediate electronic state is considered. In particular, we use a rate-equation approach to simulate cooling of ${\mathrm{SiO}}^{+}$, in which population buildup in the intermediate state is prevented by its short lifetime. We determine that Doppler cooling of ${\mathrm{SiO}}^{+}$ can be accomplished without optically repumping from the intermediate state, at the cost of causing undesirable parity flips and rotational diffusion. Since the necessary repumping would require a large number of continuous-wave lasers, optical pulse shaping of a femtosecond laser is proposed as an attractive alternative. Other candidate three-electron-level molecules are also discussed.

Figure 1

Potential energy curves generated from the molecular constants of Cai and Francois [20]. Lifetimes correspond to the net decay out of the $v=0$ level of the excited state.

Figure 2

Simplified energy level diagram for ${\mathrm{SiO}}^{+}$, omitting rotational substructure and the $A^2\Pi$ state. Electronic decay from the excited state is shown, as is vibrational decay within the ground electronic state (energy not to scale).

Figure 3

Simplified energy-level diagram for ${\mathrm{SiO}}^{+}$, omitting the rotational substructure for all electronic states and the vibrational substructure of the $X$ state. Electronic decay into each vibrational level of the $A$ state is shown along with the net decay out of the level (energy not to scale).

Figure 4

Simplified energy level diagram showing decay to the $|g_{0,J^{\prime }}^{P^{\prime }}\rangle$ states through the intermediate $|i_{v^{\prime },J^{\prime }}^{P^{\prime }}\rangle$ states (energy not to scale). Vibrational decay within the $X$ state and decay from the $|g_{v>2,J^{\prime }}^{P^{\prime }}\rangle$ states are not drawn.

Figure 5

Simplified energy level diagram showing the cooling and repumping transitions for ${\mathrm{SiO}}^{+}$ with rotational substructure and both parities included (energy not to scale). Transitions with different colors correspond to the addition of different lasers to the simulation, shown in Fig. 6.

Figure 6

Total number of photons scattered for different laser configurations. The shaded band corresponds to cooling to the Doppler limit from room temperature. The upper limit corresponds a laser detuning fixed at half the natural line width, and the lower limit corresponds to a variable detuning that follows the temperature-dependent Doppler width. Cooling (C$x$) and repumping (R$x$) wavelengths are shown in Fig. 5. From the plot, we see that we require a vibrational repump for the $v^{\prime }⩽2$ levels as well as a rotational repump for $J^{\prime }=\pm 5/2$.

Figure 7

Total number of photons scattered, treating ${\mathrm{SiO}}^{+}$ alternatively as a two-electronic-state molecule (ignoring the $A$ state) and as a three-electronic-state molecule (full treatment). The shaded band corresponds to cooling to the Doppler limit from room temperature for different laser detunings (similar to Fig. 6). Cooling (C$x$) and repumping (R$x$) wavelengths are shown in Fig. 5.