Multimode Gaussian steady state engineering in optomechanical systems with a squeezed reservoir

We investigate a theoretical protocol for the dissipative stabilization of mechanical quantum states in a multimode optomechanical system composed of multiple optical and mechanical modes. The scheme employs a single squeezed reservoir that drives one of the optical modes, while the remaining optical modes mediate an effective phonon-phonon interaction Hamiltonian. The interplay between these coherent interactions and the dissipation provided by the squeezed bath enables the steady-state preparation of targeted quantum states of the mechanical modes. In the absence of significant uncontrolled noise sources, the resulting dynamics closely approximate the model introduced in [Phys. Rev. Lett. 126, 020402 (2021)]. We analyze the performance of this protocol in generating mechanical cluster states defined on rectangular graphs.