  Optomagnonics is a new field of research in condensed matter physics and
quantum optics focused on strong magnon-photon interactions. Particular
interest concerns realistic, experimentally feasible materials and prototype
cheap elements for futuristic nanodevices implemented in the processing or
storing of quantum information. Quantifying the entanglement between two
continuous bosonic modes, such as magnons and photons, is not trivial. The
state-of-the-art for today is the logarithmic negativity, calculated through
the quantum Langevin equations subjected to thermal noise. However, due to its
complexity, this method requires further approximation. In the present work, we
propose a new procedure that avoids the linearization of dynamics. Prior
analyzing the quantum entanglement, we explore the nonlinear semiclassical
dynamics in detail and precisely define the phase space. The typical nonlinear
dynamical system holds bifurcation points and fixed points of different
characters in its phase space. Our main finding is that entanglement is not
defined in the Saddle Point region. On the other hand, the maximum of the
entanglement corresponds to the region near the border between the Stable node
and Stable spiral regions. In numerical calculations, we considered a
particular system: optomagnonic crystal based on the yttrium iron garnet (YIG)
slab with the periodic air holes drilled in the slab. In our case,
Magnon-photon interaction occurs due to the magneto-electric effect in YIG. We
provide explicit derivation of the coupling term. Besides, we calculate photon
modes for a particular geometry of the optomagnonic crystal. We analyzed the
amplitude-frequency characteristics of the optomagnonic crystal and showed that
due to the instability region, one could efficiently switch the mean magnon
numbers in the system and control entanglement in the system.
