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Cavity quantum optomechanics has emerged as a new platform for quantumscience and technology with applications ranging from quantum-informationprocessing to tests of the foundations of physics. Of crucial importance foroptomechanics is the generation and verification of non-Gaussian states ofmotion and a key outstanding challenge is the observation of a canonicaltwo-mode Schr\"odinger-cat state in the displacement of two mechanicaloscillators. In this work, we introduce a pulsed approach that utilizes thenonlinearity of the radiation-pressure interaction combined withphoton-counting measurements to generate this entangled non-Gaussian mechanicalstate, and, importantly, describe a protocol using subsequent pulsedinteractions to verify the non-Gaussian entanglement generated. Our pulsedverification protocol allows quadrature moments of the two mechanicaloscillators to be measured up to any finite order providing a toolset forexperimental characterisation of bipartite mechanical quantum states andallowing a broad range of inseparability criteria to be evaluated. Keyexperimental factors, such as optical loss and open-system dynamics, arecarefully analyzed and we show that the scheme is feasible with only minorimprovements to current experiments that operate outside the resolved-sidebandregime. Our scheme provides a new avenue for quantum experiments with entangledmechanical oscillators and offers significant potential for further researchand development that utilizes such non-Gaussian states for quantum-informationand sensing applications, and for studying the quantum-to-classical transition.

Two-mode Schr\"odinger-cat states with nonlinear optomechanics: generation and verification of non-Gaussian mechanical entanglement