  An open question in mechanics is whether mechanical resonators can be made
nonlinear with vibrations approaching the quantum ground state. This requires
engineering a mechanical nonlinearity far beyond what has been realized thus
far. Here we discovered a mechanism to boost the Duffing nonlinearity by
coupling the vibrations of a nanotube resonator to single-electron tunneling
and by operating the system in the ultrastrong coupling regime. Remarkably,
thermal vibrations become highly nonlinear when lowering the temperature. The
average vibration amplitude at the lowest temperature is 13 times the
zero-point motion, with approximately 42% of the thermal energy stored in the
anharmonic part of the potential. Our work paves the way for realizing
mechanical Schrodinger cat states [1], mechanical qubits [2, 3], and quantum
simulators emulating the electron-phonon coupling [4].
