Carbon nanotubes are excellent nano-electromechanical systems, combining high
resonance frequency, low mass, and large zero-point motion. At cryogenic
temperatures they display high mechanical quality factors. Equally they are
outstanding single electron devices with well-known quantum levels and have
been proposed for the implementation of charge or spin qubits. The integration
of these devices into microwave optomechanical circuits is however hindered by
a mismatch of scales, between typical microwave wavelengths, nanotube segment
lengths, and nanotube deflections. As experimentally demonstrated recently in
[Blien et al., Nat. Comm. 11, 1363 (2020)], coupling enhancement via the
quantum capacitance allows to circumvent this restriction. Here we extend the
discussion of this experiment. We present the subsystems of the device and
their interactions in detail. An alternative approach to the optomechanical
coupling is presented, allowing to estimate the mechanical zero point motion
scale. Further, the mechanical damping is discussed, hinting at hitherto
unknown interaction mechanisms.Comment: 17 pages, 13 figures, 3 table