Phonon condensation and cooling via nonlinear feedback

We propose a method to control the energy distribution in multimode mechanical systems using a single nonlinear feedback loop. We demonstrate that this feedback mechanism simultaneously amplifies the fundamental vibrational mode while suppressing all higher-order modes, effectively channeling energy into the lowest-frequency mode. This process mimics the energy redistribution of Fr\"{o}hlich condensation but is achieved here through a designed feedback force that combines a ``low-pass gain'' and a ``high-pass loss''. In the feedback-induced steady state, the fundamental mode exhibits a phase-space distribution similar to that of a phonon laser, characterized by a ring shape and amplitude squeezing. Additionally, we show that the linewidth of the fundamental mode is narrowed by an order of magnitude, corresponding to a significant enhancement in phase coherence. This scheme offers a robust approach to generating coherent mechanical states and phonon lasing without the need for optical gain media or intrinsic material nonlinearities.