Multi-order Nonlinearities and Resulting Coherent Oscillations of the States in Quantum Dot-Mechanical Resonator Hybrid System

The carbon nanotube (CNT) hybrid device composed of a gate-defined quantum dot embedded into a mechanical resonator is investigated under strong actuation condition. The Coulomb peaks are driven to the synchronous and large oscillation by the nonlinear mechanical vibration, and can be coherently modulated to the single-electron shuttle mode. Conversely, the mediated single-electron tunnelling current provides an in situ and sensitive characterization strategy for the multi-order mechanical nonlinear resonance. We show that, under the strong modulation applied via gate voltage, the swing and trampoline modes exhibit different nonlinearity behaviour, depending on both the quadratic and cubic coefficients. The nonlinearities in this hybrid system also give rise to coupling modes of quadratic and cubic resonances at combined frequencies and even fractional frequencies. Especially, the subharmonic resonance due to quadratic nonlinearity is predicted to enable the excitation of the coherent coupling state between two modes. Our theoretical mode of the intra- and intermodal nonlinearities shows a good agreement of our experimental observations quantitatively.