Robust dynamical and steady-state entanglement of phonons in the magnetic microsphere

In the field of quantum information, robust entanglement has received widespread attention. In this work, we investigate the magnon-mediated robust entanglement between the acoustic phonon and the center-of-mass motion (CMM) phonon in a yttrium iron garnet sphere. By analyzing the system's dynamic evolution, we demonstrate that phonon-phonon entanglement, which is robust with respect to the magnon mode, can be generated at a specific moment. Furthermore, we introduce a scheme for generating stable bipartite entanglement that remains robust against varying initial states by utilizing an engineered reservoir. This approach maintains effectiveness even under high dissipation rates of the magnon mode, leveraging the magnon dissipation as a beneficial resource. The approaches we present for generating robust phonon-phonon entanglement not only offer a feasible method for monitoring internal acoustic modes within magnetic systems using easily detectable CMM phonons, but also provide a viable platform for the manipulation and processing of quantum information.