  We study the dissipative quantum Duffing oscillator in the deep quantum
regime with two different approaches: The first is based on the exact Floquet
states of the linear oscillator and the nonlinearity is treated perturbatively.
It well describes the nonlinear oscillator dynamics away from resonance. The
second, in contrast, is applicable at and in the vicinity of a N-photon
resonance and it exploits quasi-degenerate perturbation theory for the
nonlinear oscillator in Floquet space. It is perturbative both in driving and
nonlinearity. A combination of both approaches yields the possibility to cover
the whole range of driving frequencies. As an example we discuss the
dissipative dynamics of the Duffing oscillator near and at the one-photon
resonance.
