The prospects of sympathetic cooling of polar molecules with magnetically cotrapped alkali-metal atoms are generally considered poor due to strongly anisotropic atom-molecule interactions leading to large spin relaxation rates. Using rigorous quantum scattering calculations based on ab initio interaction potentials, we show that inelastic spin relaxation in low-temperature collisions of CaH(${}^{2}\ensuremath{\Sigma}$) molecules with Li and Mg atoms occurs at a slow rate despite the strongly anisotropic interactions. This unexpected result, which we rationalize using multichannel quantum-defect theory, opens up the possibility of sympathetic cooling of polar ${}^{2}\ensuremath{\Sigma}$ molecules with alkali-metal atoms in a magnetic trap and with alkaline-earth-metal atoms in an optical dipole trap.
We model sympathetic cooling of ground-state CaF molecules by ultracold Li and Rb atoms. The molecules are moving in a microwave trap, while the atoms are trapped magnetically. We calculate the differential elastic cross sections for CaF-Li and CaF-Rb collisions, using model Lennard-Jones potentials adjusted to give typical values for the
s-wave scattering length. Together with trajectory calculations, these differential cross sections are used to simulate the cooling of the molecules, the heating of the atoms, and the loss of atoms from the trap. We show that a hard-sphere collision model based on an energy-dependent momentum transport cross section accurately predicts the molecule cooling rate but underestimates the rates of atom heating and loss. Our simulations suggest that Rb is a more effective coolant than Li for ground-state molecules, and that the cooling dynamics is less sensitive to the exact value of the
s-wave scattering length when Rb is used. Using realistic experimental parameters, we find that molecules can be sympathetically cooled to $100\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{K}$ in about 10 s. By applying evaporative cooling to the atoms, the cooling rate can be increased and the final temperature of the molecules can be reduced to $1 \ensuremath{\mu}\mathrm{K}$ within 30 s.