  Josephson transport through a single molecule or carbon nanotube is
considered in the presence of a local vibrational mode coupled to the
electronic charge. The ground-state solution is obtained exactly in the limit
of a large superconducting gap, and is extended to the general case by
variational analysis. Coherent charge fluctuations are entangled with
non-classical phonon states. The Josephson current induces squeezing of the
phonon mode, which is controlled by the superconducting phase difference and by
the junction asymmetry. Optical probes of non-classical phonon states are
briefly discussed.
