**Mikhail Fistul
**

I report a theoretical analysis of quantum transitions of an ac driven nonlinear oscillator. In a classical regime such a system displays a hysteresis in the amplitude of forced vibrations as a function of the ac bias amplitude. In a quantum regime a nonlinear oscillator shows a set of discrete energy levels becoming more dense as the oscillator energy increases. The presence of a resonant ac force leads to coherent quantum oscillations between the ground state and a particular excited energy level. The excited state is determined by the condition of multi-photon excitation. The frequency of quantum oscillations that can be considered as a generic Rabi frequency, depends on both the amplitude of ac drive and nonlinearity of a system. We obtain that the multi-photon Rabi frequency is strongly enhanced as the ac amplitude exceeds the critical value.The quantum oscillations manifest themselves in a sharp dip of the amplitude of forced oscillations as a function of ac bias frequency and/or in an enhancement of particle escape from a potential well. We discuss also as a specific source of decoherence, i.e. a low frequency noise, destroys the quantum oscillations. The coherent quantum oscillations can be observed in various systems, e.g. Josephson junctions and nanomechanical resonators, molecular magnets or vibrational spectrum of molecules.