  We investigate the Josephson current through a suspended carbon nanotube
double quantum dot which, at sufficiently low temperatures, is characterized by
the ground state of the electronic subsystem. Depending on parameters like a
magnetic field or the inter-dot coupling, the ground state can either be a
current-carrying singlet or doublet, or a blockaded triplet state. Since the
electron-vibration interaction has been demonstrated to be electrostatically
tuneable, we study in particular its effect on the current-phase relation. We
show that the coupling to the vibration mode can lift the current-suppressing
triplet blockade by inducing a quantum phase transition to a ground state of a
different total spin. Our key finding is the development of a triple point in
the Josephson current parameterized by the resonator coupling and the Josephson
phase. The quantum phase transitions around the triple point are directly
accessible through the critical current and resilient to moderately finite
temperatures. The proposed setup makes the mechanical degree of freedom part of
a superconducting hybrid device which is interesting for ultra-sensitive
displacement detectors.
