We demonstrate sympathetic sideband cooling of a $^{ 40}$CaH$^{ +}$ molecular ion co-trapped with a $^{ 40}$Ca$^{ +}$ atomic ion in a linear Paul trap. Both axial modes of the two-ion chain are simultaneously cooled to near the ground state of motion. The center of mass mode is cooled to an average quanta of harmonic motion $\overline{ n}_{ \mathrm{ COM}} = 0.13 \pm 0.03$, corresponding to a temperature of $12.47 \pm 0.03 ~\mu$K. The breathing mode is cooled to $\overline{ n}_{ \mathrm{ BM}} = 0.05 \pm 0.02$, corresponding to a temperature of $15.36 \pm 0.01~\mu$K.
Categories: physics.atom-ph