Manipulating the long-range strong photon-magnon coupling via a saturable gain

Long-range tunable strongly coupled photon-magnon systems are crucial for large-scale hybrid networks enabling coherent information processing. Here, we experimentally fabricate a photon-magnon system with a saturable gain, achieving long-range strong coupling in the linear regime for the first time, aside from the nonlinear regime. By modulating the propagation phase of traveling waves mediating photons and magnons, we achieve flexible switching between coherent coupling and dissipative coupling. Theoretically, we construct a Hamiltonian model comprising a van der Pol oscillator and a harmonic oscillator, from which we derive the dynamical equations of the cavity magnonic system and numerically simulate the time evolution of photon and magnon modes amplitudes under the long-range strong coupling. The results reveal energy exchange in the linear regime and mode synchronization in the nonlinear regime. Our work integrates experimental results and theoretical models, establishing a foundation for cross-platform information exchange and controllable coupling in hybrid systems.