  Cavity optomechanics allows the characterization of a vibration mode, its
cooling and quantum manipulation using electromagnetic fields. Regarding
nanomechanical as well as electronic properties, single wall carbon nanotubes
are a prototypical experimental system. At cryogenic temperatures, as high
quality factor vibrational resonators, they display strong interaction between
motion and single-electron tunneling. Here, we demonstrate large optomechanical
coupling of a suspended carbon nanotube quantum dot and a microwave cavity,
amplified by several orders of magnitude via the nonlinearity of Coulomb
blockade. From an optomechanically induced transparency (OMIT) experiment, we
obtain a single photon coupling of up to $g_0=2\pi\cdot 95\,\textrm{Hz}$. This
indicates that normal mode splitting and full optomechanical control of the
carbon nanotube vibration in the quantum limit is reachable in the near future.
Mechanical manipulation and characterization via the microwave field can be
complemented by the manifold physics of quantum-confined single electron
devices.
