  We investigate the nonlinear response of a vibrating suspended nanomechanical
beam on external periodic driving. The amplitude of the fundamental transverse
mode behaves thereby like a weakly damped quantum particle in a driven
anharmonic potential. Upon using a Born-Markovian master equation, we calculate
the fundamental mode amplitude for varying driving frequencies. In the
nonlinear regime, we observe resonances which are absent in the corresponding
classical model. They are shown to be associated with resonant multi-phonon
excitations.
  Furthermore, we identify resonant tunneling in a dynamically induced bistable
effective potential.
