Driving collective RPA modes by a time-dependent Dyson map

We study a time-dependent non-Hermitian generalisation of the Sch\"utte-Da~Provid\^encia model describing a bosonic mode coupled to collective particle-hole excitations. Using a time-dependent Dyson map, we construct a Hermitian counterpart and reduce the collective fermionic sector by means of the random phase approximation (RPA). The resulting dynamics is mapped to two time-dependent harmonic-oscillator branches with instantaneous RPA frequencies $W_\pm(t)$. We determine the corresponding stability regions and compute transition probabilities between instantaneous oscillator states. In first-order instantaneous-basis perturbation theory the leading transition $n\to n+2$ is proportional to $\dot W_j/W_j$, showing that it is purely nonadiabatic and absent in the time-independent case. We compare this result with exact Lewis-Riesenfeld transition amplitudes within the RPA approximation. Numerical examples show that different components of the Dyson map provide distinct driving mechanisms: the scaling parameter modulates the effective coupling, while the squeezing parameter acts through a moving-boundary contribution. In both cases the induced collective transitions exhibit parametric-resonance peaks and sideband structures.