Optomechanical quantum entanglement mediated by acoustic phonon fields

We present exact solutions for the quantum time evolution of two spatially separated, local inductor-capacitor (LC) oscillators that are coupled optomechanically to a long elastic strip that functions as a quantum thermal acoustic field environment. We show that the optomechanical coupling to the acoustic environment gives rise to causal entanglement dynamics between the two LC oscillators in the absence of resonant photon exchange between them, and that significant entanglement develops regardless of the environment temperature. Such a process establishes that distributed entanglement may be generated between superconducting qubits via a connected phonon bus bar, without the need for resonant phonon release and capture.