Quantum representations slow mode states

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

Appendix N shows that within quantum representation /// and according to Eq. (N12), when the fast mode is in its first excited state, the ground state of the slow mode is a coherent state, that is, an eigenstate of the lowering operator ... 

As a consequence of these equations, it appears that, when the fast mode is in its ground state, the slow mode does not involve any dynamics that might be the X quantum representation. 

In quantum representation ///, the density operator of the slow mode corresponding to the situation where the fast mode is in its ground state 0 ) is unchanged with respect to representation // ... 

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