Fluctuation instabilities via internal resonance in a multimode membrane as a mechanism for frequency combs

We explore self-induced parametric coupling, also called internal resonances (IRs), in a membrane nanoelectromechanical system. Specifically, we focus on the formation of a limit cycle manifesting as a phononic frequency comb. Utilizing a pump-noisy-probe technique and theoretical modeling, we reveal the behavior of mechanical excitations revealing themselves as sidebands of the stationary IR response. We find that when the energy-absorbing excitation of a lower mode is parametrically-upconverted to hybridize with a higher mode, significant squeezing and bimodality in the upper mode occurs. Instead, when the upconverted absorbing excitation hybridizes with an emitting sideband of the higher mode, a Hopf bifurcation occurs and a limit cycle forms, manifesting as a frequency comb. We thus reveal a unique mechanism to obtain frequency combs in parametrically-coupled modes. We furthermore demonstrate a rich variety of IR effects, the origin of which significantly extends beyond standard linear parametric coupling phenomena. Our findings enhance the understanding of energy transfer mechanisms with implications for advanced sensing technologies and novel phononic metamaterials.