  Recent advances have attracted attention to non-standard Josephson junctions
in which a supercurrent can flow despite zero phase difference between the
constituent superconducting leads. Here, we propose a zero-phase-difference
nanoelectromechanical junction which, in contrast to other considered systems,
exhibits symmetry between leftward and rightward tunneling through the
junction. We show that a supercurrent is nevertheless possible as a result of
spontaneous symmetry breaking. In the suggested junction, the supercurrent is
mediated by tunneling via a superconducting Cooper-pair box on a bendable
cantilever. An alternating electric potential parametrically excites mechanical
oscillations which are synchronized with charge oscillations of the box. This
leads to coherent transfer of Cooper pairs through the junction. The direction
of the supercurrent is a result of spontaneous symmetry breaking and thus it
can be reversed without changing the parameters.
