Qubit-mediated time-robust entangling of oscillators in thermal environments

We consider two separated oscillators initially in equilibrium and continuously interacting with thermal environments and propose a way to entangle them using a mediating qubit. An appropriate interaction allows for an analytic treatment of the open system, removes the necessity of fine-tuning interaction times, and results in a high tolerance of the entanglement to finite temperature. The entanglement thus produced between the oscillators can be verified either through a Bell inequality relying on oscillator parity measurements or through conditional extraction of the entanglement on two mutually noninteracting qubits. The latter process also shows that the generated mixed-entangled state of the oscillators is an useful resource for entangling qubits. By allowing for influences from environments, taking feasible qubit-oscillator interactions and measurement settings, this scheme should be implementable in a variety of experimental setups. The method presented for the solution of the master equation can also be adapted to a variety of problems involving the same form of qubit-oscillator interaction