  We demonstrate a quantum regime of dissipative nonlinear oscillator where
creation of Fock states as well as superpositions of Fock states are realized
for time-intervals exceeding the characteristic decoherence time. Preparation
of quantum states is conditioned by strong Kerr nonlinearity as well as by
excitation of resolved lower oscillatory energy levels with specific train of
Gaussian pulses. This provides practical signatures to look for in experiments
with cooled nonlinear oscillators.
