  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.
