A biased yttrium iron garnet (YIG) sphere with a uniform precession Kittel mode can also exhibit a quantized higher-order magnetostatic (HMS) mode due to additional inhomogeneities in the magnetic field. This phenomenon results in a multimode system within a single structure. Employing this two-mode YIG in a microwave cavity leads to a tripartite system that is easier to control compared to a conventional cavity with two YIG spheres. Dynamics of photon and magnon blockade in a cavity magnonic system consisting of a YIG sphere with Kittel and HMS modes are investigated. In this study, we take into account the presence of self-Kerr effects for both Kittel and HMS modes, as well as the cross-Kerr phenomenon between them. The impact of self-Kerr and cross-Kerr parameters on photon and magnon blockade is examined. Finally, we calculate optimal practical parameters for this system to set up simultaneous photon and magnon blockade.

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