Sympathetic cooling of molecular ion motion to the ground state

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 ${\bar{n}}_{\mathrm{C}\mathrm{O}\mathrm{M}}=0.13\pm 0.03$, corresponding to a temperature of $12.47\pm 0.03\mu$K. The breathing mode is cooled to ${\bar{n}}_{\mathrm{B}\mathrm{M}}=0.05\pm 0.02$, corresponding to a temperature of $15.36\pm 0.01\mu$K.