Implementing a cross-Kerr interaction between microwave and optical cavities and its application in generating a hybrid continuous-variable-discrete-variable entangled state

Microwave-optical interactions and hybrid entangled states are crucial for building hybrid quantum networks. In this work, we propose an approach for realizing a cross-Kerr nonlinear interaction between microwave and optical cavities using an ensemble of nitrogen-vacancy centers (NV ensemble). This cross-Kerr interaction is achieved by an NV ensemble dispersively coupled to a microwave cavity and an optical cavity. As an application, we show that the cross-Kerr interaction can be used to create a hybrid entangled state of a discrete-variable (DV) optical qubit and a continuous-variable (CV) optical qubit. The DV optical qubit here refers to a qubit, with two logic states encoded via the vacuum and single-photon states of the optical cavity. The CV optical qubit means a qubit, whose two logic states are encoded through two quasi-orthogonal coherent states or cat states of the microwave cavity. Our method is quite simple because it only requires a single-step operation. Numerical simulations demonstrate that high-fidelity generation of the proposed hybrid state is feasible within current experimental technology. This proposal is universal and the NV ensemble can be replaced by other quantum transducers, such as magnons, rare-earth-doped crystals, and silicon-vacancy centers.