Entanglement and steering in a three-dimensional cavity magnomechanics system

We investigate steady-state entanglement and steering in a three-dimensional cavity magnomechanics system. From the system's equations of motion, we derive the linearized Lyapunov equation for steady states, introduce logarithmic negativity to quantify entanglement, and apply a quantization method for Gaussian states with continuous variables to measure steering. Numerical simulations reveal that steady-state entanglement and steering can be established between various system components---such as the two cavity modes, the cavity mode and magnon mode, and the cavity mode and phonon mode---by selecting experimentally feasible detunings, dissipation rates, and coupling rates. We develop a three-dimensional cross-cavity model to study bipartite and tripartite entanglement and explore parameter adjustments to achieve both one-way and two-way steering. Additionally, we confirm the system's strong robustness to temperature variations. This work provides a platform for studying entanglement and steering beyond squeezed optical drives and has potential applications in quantum information processing and quantum sensing.