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.

Figure 1

(a) and (b) Schematic representation of different types of entangled Schrödinger cat states of two suspended masses moving harmonically along the $x$ axis. Here $\alpha$ is the displacement amplitude from the equilibrium position. (c) Sketch of a possible experimental implementation with a quantum electromechanical circuit.

Figure 2

(a) Periodic revival of the concurrence, undergoing decoherence, signifies the entangled superposition between the masses for small (dashed line) and large (solid line) displacement amplitudes. For larger amplitudes the decay of entanglement revival under decoherence occurs with a higher rate. (b) Preparation of large entangled superposition using the ${\left[\pi \right]}_{g\leftrightarrow e}$ pulse sequence and monitoring the concurrence afterwards without decoherence (solid line) and with decoherence (dashed line).

Figure 3

(a) Time evolution of entanglement of two qubits in terms of concurrence with (dashed line) and without (solid line) decoherence and (b) probability of finding the $A$ qubit in its ground state and the $B$ qubit in its first excited state. This quantity is obtained by simulating the protocol and the evolution of the prepared state using the parameters listed in Table 1.