  The coupling of superconducting systems to mechanical resonators is an
emerging field, with wide reaching implications including high precision
sensing and metrology. Experimental signatures of this coupling have so far
been small, seldom and often reliant on high frequency AC electronics. To
overcome this limitation, in this work we consider a mechanical resonator
suspended between two superconducting contacts to form a suspended Josephson
junction in which the electronic normal- and super-currents can be coupled to
mechanical motion via the Lorentz force due to an external magnetic field. We
show both analytically and numerically that this electro-mechanical coupling
produces unprecedented mechanically-induced hysteresis loops in the junction's
DC I-V characteristic. Firstly, we unveil how this new hysteresis may be
exploited to access a huge mechanically-induced Shapiro-like voltage plateau,
extending over a current range comparable with the junction's critical current.
We then investigate a sudden mechanically-induced retrapping that occurs at
strong coupling. Our analytical treatment provides a clear explanation for the
effects above and allows us to derive simple relationships between the features
in the DC I-V characteristic and the resonance frequency and quality factor of
the mechanical resonator. We stress that our setup requires only DC current
bias and voltage measurements, allowing the activation and detection of
high-frequency mechanical oscillations in state of the art devices and without
the need of any AC equipment.
