  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.
