Resonance and Transition-Channel Competition in Cavity-Assisted Floquet-State Stabilization

Static coupling to a dissipative cavity has been proposed as a route to autonomously stabilize Floquet eigenstates. The cavity provides an energy-selective relaxation channel: when its photon energy matches a quasienergy difference modulo drive quanta, it mediates transitions between Floquet states, followed by irreversible photon loss. Here, we show that at intermediate driving frequencies, this resonance condition alone is not sufficient for stabilization. Instead, the steady state is controlled by the interplay between resonance selection and the cavity-induced transition strengths between Floquet states, which can either cooperate or compete. This interplay leads to successful stabilization, mixed steady states despite resonance, and complete breakdown near Floquet-zone hybridization. We further show that the same mechanism extends to multilevel spin-S systems, providing a general framework for understanding when static dissipation can reliably stabilize driven quantum states.