Quantum coupled oscillators versus forced oscillator

We compare the time evolutions of two quantum systems: a quantum harmonic o scillator driven by a classical force and two coupled quantum harmonic osci llators. We find the conditions under which the first system can be conside red as a 'mean-field approximation' to the second one. This happens in the weak coupling limit provided that the 'master' oscillator was initially in a highly excited quantum state with small fluctuations of the canonical var iables (such as the coherent state or displaced thermal state), while the ' subordinate' oscillator must not be 'too excited'. The essential difference between the resonance and nonresonance cases is discussed. In particular, the effect of the resonance-quantum-state exchange is demonstrated.