Entanglement and steering in a cross-shaped double-cavity with a magnetic sphere and driven by a squeezed vacuum field

We have explored steady-state entanglement and steering in a cross-shaped double-cavity with a levitating magnetic sphere and driven by a squeezed vacuum field. From the equations of motion of the system, we derived the linearized Lyapunov equation for the steady-states, introduced the logarithmic negativity to measure the entanglement and a similar quantity to measure the steering, and conducted calculations in parameter regions around stable steady-states. Numerical results show that steady-state entanglement and steering between various components of the system, such as cavity-cavity, cavity-magnon, and cavity-phonon, can form by choosing experimentally feasible detunings, dissipation rates, and coupling rates. Moreover, one-way steering can be achieved through appropriately adjusting the system parameters. The proposed model provides a platform for the study of the entanglement and one-way steering, and may find itself applications in quantum information processing and quantum cryptography.