The realization of multimode optomechanical interactions in the single-photon strong-coupling regime is a desired task in cavity optomechanics, but it remains a challenge in realistic physical systems. In this work, we propose a reliable scheme to simulate a three-mode optomechanical system working in the single-photon strong-coupling regime based on the Fredkin-type interaction. This is achieved by utilizing two strong drivings to the two exchangly-coupled modes in the Fredkin-type coupling involving one optical mode and two mechanical-like modes. As an application of this enhanced three-mode nonlinear optomechanical coupling, we show how to generate entangled-cat states of the mechanical-like modes using the conditional displacement mechanism. The quantum coherence effects in the generated states are investigated by calculating two-mode joint Wigner function and quantum entanglement. The influence of the dissipation effect on the state generation is considered in the open-system case.
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