Fig. 1: Schematic diagram of the experimental setup.

a The microwave signal generated by a vector network analyzer (VNA) is equally divided into two beams through a power splitter (PS). The amplitude and phase of each beam can be independently tuned by a variable attenuator (VA) and variable phase shifter (VPS). The two microwave fields are injected into the cavity after through two directional couplers (DCs) connected to the two ports, and the output fields are measured by the VNA after the DCs. The microwave cavity is a 3D oxygen-free copper cavity with dimensions of 60 × 25 × 6 mm³, and a 0.25-mm-diameter YIG sphere is placed at the antinode of the magnetic field of the cavity TE₁₀₂ mode. The YIG sphere is supported by a horizontal glass capillary and can move freely to reduce the mechanical damping. At the top of the cavity, a coil connected with a microwave source is used to directly drive the YIG sphere. Each cavity port is fixed with a SMA adaptor, and the length of the SMA adaptor’s pin entering the cavity δl₁₍₂₎ can be tuned. The external magnetic field B₀ (in the z direction) is applied to set the frequency of the magnon mode. The experiment is performed at room temperature. b Three interacting modes of the system, including a cavity mode, a magnon mode, and a vibration mode. At the top is the magnetic field distribution of the cavity TE₁₀₂ mode.