Tunable nonlinear electromechanics at the zero-point motion scale

Nonlinearity at the scale of zero-point motion opens new possibilities for the control and readout of nanomechanical systems, but achieving this remains a formidable challenge. Here we demonstrate that ultrastrong coupling (USC) between a nanotube mechanical oscillator and a double-quantum-dot electronic two-level system enables a mechanical Kerr (Duffing) nonlinearity at the zero-point motion scale. In the dispersive regime, this large coupling yields a mechanical anharmonicity of $\alpha = 1.4\%$ - three orders of magnitude larger than in previous work - while preserving the predominantly mechanical nature of the lowest energy states. We further demonstrate a purely quadratic cavity-based continuous readout of the mechanical motion. This continuous nonlinear optomechanical readout is enforced by a double-quantum dot symmetry, which can be broken by gate tuning to introduce a large linear transduction. These results establish a tunable USC platform that enables strong mechanical anharmonicity and nonlinear continuous readout at the zero-point motion scale.