Reprint of : Dynamics of coupled vibration modes in a quantum non-linear mechanical resonator

• If two flexural modes are mechanically coupled and one of them is driven the other mode responds.
• The response is similar to the parametric resonance in the classical physics.
• In the quantum case, the non-driven mode responds even below the classical threshold.
• The magnitude of the response is comparable to the amplitude of the zero-point motion.

We investigate the behaviour of two non-linearly coupled flexural modes of a doubly clamped suspended beam (nanomechanical resonator). One of the modes is externally driven. We demonstrate that classically, the behavior of the non-driven mode is reminiscent of that of a parametrically driven linear oscillator: it exhibits a threshold behavior, with the amplitude of this mode below the threshold being exactly zero. Quantum-mechanically, we were able to access the dynamics of this mode below the classical parametric threshold. We show that whereas the mean displacement of this mode is still zero, the mean squared displacement is finite and at the threshold corresponds to the occupation number of 1/2. This finite displacement of the non-driven mode can serve as an experimentally verifiable quantum signature of quantum motion.