Abstract
We theoretically study a multimode nonequilibrium optomechanical system coupled to squeezed-vacuum and thermal reservoirs, focusing on the interplay between quantum coherence, dissipation, and heat transport. Using covariance-matrix methods, we analyze bipartite entanglement, EPR steering, and energy currents in the steady state. We show that weak optomechanical coupling preserves squeezing-induced entanglement, whereas strong coupling enhances phonon-mediated decoherence and suppresses quantum correlations. The system exhibits one-way quantum steering, indicating directional quantum information transfer. We further demonstrate that reservoir squeezing acts as a nonclassical energy source that sustains nonequilibrium heat flow, while substrate-mediated phonon correlations generate geometry-dependent thermal rectification controlled by the separation between mechanical resonators. Interference between transport pathways can strongly suppress heat currents, analogous to dark-state effects in quantum optics. These findings provide a unified framework linking squeezing, entanglement, steering, and heat transport, with potential applications in quantum thermal devices, phononic routing, and heat-resilient quantum technologies.