Heralded entangled coherent states between spatially separated massive resonators

We put forward an experimentally feasible scheme for heralded entanglement generation between two distant macroscopic mechanical resonators. The protocol exploits a hybrid quantum device, a qubit interacting with a mechanical resonator as well as a cavity mode, for each party. The cavity modes interfere on a beam splitter followed by suitable heralding detections, which postselect a hybrid entangled state with success probability 1/2. Subsequently, by local measurements on the qubits, a mechanically entangled coherent state can be achieved. The mechanical entanglement can be further verified via monitoring the entanglement of the qubit pair. The setup is envisioned as a test bed for sensing gravitational effects on the quantum dynamics of gravitationally coupled massive objects. As a concrete example, we illustrate the implementation of our protocol using the current circuit QED architectures.