  The dissipative quantum dynamics of an anharmonic oscillator is investigated
theoretically in the context of carbon-based nano-mechanical systems. In the
short-time limit, it is known that macroscopic superposition states appear for
such oscillators. In the long-time limit, single and multi-phonon dissipation
lead to decoherence of the non-classical states. However, at zero temperature,
as a result of two-phonon losses the quantum oscillator eventually evolves into
a non-classical steady state. The relaxation of this state due to thermal
excitations and one-phonon losses is numerically and analytically studied. The
possibility of verifying the occurrence of the non-classical state is
investigated and signatures of the quantum features arising in a ring-down
setup are presented. The feasibility of the verification scheme is discussed in
the context of quantum nano-mechanical systems.
