  We present a study of the controllable nonlinear dynamics of a
micromechanical beam coupled to a dc-SQUID (superconducting quantum
interference device). The coupling between these systems places the modes of
the beam in a highly nonlinear potential, whose shape can be altered by varying
the control parameters of the SQUID. We detect the position of the beam by
placing it in an optical cavity, which frees the SQUID to be used solely for
actuation. This enables us to probe the previously unexplored full parameter
space of this device. We measure the frequency response of the beam and find
that it displays a periodic dependence on applied magnetic flux. To account for
this, we develop a model based on the standard theory for SQUID dynamics. In
addition, with the aim of understanding if the device can reach nonlinearity at
the single phonon level, we use this model to show that the responsivity of the
current circulating in the SQUID to the position of the beam can become
divergent, with its magnitude limited only by noise. This suggests a direction
for the generation of macroscopically distinguishable superposition states of
the beam.
