  We present a scheme leading to ground-state cooling of the fundamental
out-of-plane (flexural) mode of a suspended graphene sheet. Our proposal
exploits the coupling between a driven Rydberg atom and the graphene resonator,
which is enabled by vacuum forces. Thanks to the large atomic polarizability of
the Rydberg states, the Casimir-Polder force is several orders of magnitude
larger than the corresponding force achieved for atoms in the ground state. By
playing with the distance between the atom and the graphene membrane, we show
that resolved sideband cooling is possible, bringing the occupation number of
the fundamental flexural mode down to its quantum limit. Our findings are
expected to motivate physical applications of graphene at extremely low
temperatures.
