  We demonstrate that a suspended nanowire forming a weak link between two
superconductors can be cooled to its motional ground state by a supercurrent
flow. The predicted cooling mechanism has its origins in magnetic field induced
inelastic tunneling of the macroscopic superconducting phase associated with
the junction. Furthermore, we show the voltage-drop over the junction is
proportional to the average population of the vibrational modes in the
stationary regime, a phenomena which can be used to probe the level of cooling.
