Conventional vs gauge invariant quantum transition probabilities in simple systems

The evaluation of gauge-invariant quantum transition probabilities for systems stimulated by external electromagnetic fields in general poses the problem of using a proper basis-defining hamiltonian operator, depending on the gauge chosen for introducing the effect of the external fields in the theoretical framework. A missed understanding of this point can lead to inconsistences, examples of which are presented and discussed for the case of a simple harmonic oscillator linearly driven by a time-dependent force. The behaviour of a quantum oscillator subjected (a) to a monochromatic field suddenly switched-on at a given time, (b) to a (short-duration) pulse is investigated exactly at high intensities and time-resolved gauge-invariant vs. ''conventional'' (i.e., generally false) transition probabilities are compared. In addition, approximation for these transition probabilities in terms of truncated bases of free-field harmonic oscillator eigenstates are derived and compared with the exact results