  Cavity quantum optomechanics has emerged as a new platform for quantum
science and technology with applications ranging from quantum-information
processing to tests of the foundations of physics. Of crucial importance for
optomechanics is the generation and verification of non-Gaussian states of
motion and a key outstanding challenge is the observation of a canonical
two-mode Schr\"odinger-cat state in the displacement of two mechanical
oscillators. In this work, we introduce a pulsed approach that utilizes the
nonlinearity of the radiation-pressure interaction combined with
photon-counting measurements to generate this entangled non-Gaussian mechanical
state, and, importantly, describe a protocol using subsequent pulsed
interactions to verify the non-Gaussian entanglement generated. Our pulsed
verification protocol allows quadrature moments of the two mechanical
oscillators to be measured up to any finite order providing a toolset for
experimental characterisation of bipartite mechanical quantum states and
allowing a broad range of inseparability criteria to be evaluated. Key
experimental factors, such as optical loss and open-system dynamics, are
carefully analyzed and we show that the scheme is feasible with only minor
improvements to current experiments that operate outside the resolved-sideband
regime. Our scheme provides a new avenue for quantum experiments with entangled
mechanical oscillators and offers significant potential for further research
and development that utilizes such non-Gaussian states for quantum-information
and sensing applications, and for studying the quantum-to-classical transition.
