  We investigate the sympathetic relaxation of a free-standing, vibrating
carbon nano-tube that is mounted on an atom chip and is immersed in a cloud of
ultra-cold atoms. Gas atoms colliding with the nano-tube excite phonons via a
Casimir-Polder potential. We use Fermi's Golden Rule to estimate the relaxation
rates for relevant experimental parameters and develop a fully dynamic theory
of relaxation for the multi-mode phononic field embedded in a thermal atomic
reservoir. Based on currently available experimental data, we identify the
relaxation rates as a function of atom density and temperature that are
required for sympathetic ground state cooling of carbon nano-tubes.
