  When there is a certain amount of field inhomogeneity, the biased
ferrimagnetic crystal will exhibit the higher-order magnetostatic (HMS) mode in
addition to the uniform-precession Kittel mode. In cavity magnonics, we show
both experimentally and theoretically the cross-Kerr-type interaction between
the Kittel mode and HMS mode. When the Kittel mode is driven to generate a
certain number of excitations, the HMS mode displays a corresponding frequency
shift and vice versa. The cross-Kerr effect is caused by an exchange
interaction between these two spin-wave modes. Utilizing the cross-Kerr effect,
we realize and integrate a multi-mode cavity magnonic system with only one
yttrium iron garnet (YIG) sphere. Our results will bring new methods to
magnetization dynamics studies and pave a way for novel cavity magnonic devices
by including the magnetostatic mode-mode interaction as an operational degree
of freedom.
