Parametric instability in a magnomechanical system

We study parametric instability in a magnomechanical system, specifically examining magnon tunneling between moving ferromagnetic insulators. Our analysis reveals that quantum fluctuations generate spin currents above a critical velocity threshold, while no spin currents occur below this threshold at low temperatures. The critical velocity depends on magnon stiffness and Zeeman energy. Approaching the threshold, the spin current becomes divergent, linked to the $PT$-symmetry-breaking transition. This enhanced behavior could offer high-sensitivity measurements and efficient spin current generation in magnon-based quantum technology.