Trapping and Ground-State Cooling of a Single H${}_2^{+}$

We demonstrate co-trapping and sideband cooling of a H${}_2^{+}$ - ${}^9$Be${}^{+}$ ion pair in a cryogenic Paul trap. We study the chemical lifetime of H${}_2^{+}$ and its dependence on the apparatus temperature, achieving lifetimes of up to $11_{-3}^{+6}$ h at 10 K. We demonstrate cooling of two of the modes of translational motion to an average phonon number of 0.07(1) and 0.05(1), corresponding to a temperature of 22(1) $\mu$K and 55(3) $\mu$K respectively. Our results provide a basis for quantum logic spectroscopy experiments of H${}_2^{+}$, as well as other light ions such as HD${}^{+}$, H${}_3^{+}$, and He${}^{+}$.