  We consider the problem of understanding the basic features displayed by
quantum systems described by parametric oscillators whose time-dependent
frequency parameter $\omega(t)$ varies continuously during evolution so to
realise quenching protocols of different types. To this scope we focus on the
case where $\omega(t)^2$ behaves like a Morse potential, up to possible sign
reversion and translations in the $(t,\omega^2)$ plane. We derive closed form
solution for the time-dependent amplitude of quasi-normal modes, which is the
very fundamental dynamical object entering the description of both classical
and quantum parametric oscillators, and highlight its significant
characteristics for distinctive cases arising based on the driving specifics.
After doing so, we provide an insight on the way quantum states evolve by
paying attention on the position-momentum Heisenberg uncertainty principle and
the statistical aspects implied by second-order correlation functions over
number-type states.
