  We investigate quantum interference effects in a superconducting Cooper-pair
box by taking into account the possibility of tunneling processes involving one
and two Cooper pairs. The quantum dynamics is analysed in a framework of
three-level model. We compute Landau-Zener probabilities for a linear sweep of
the gate charge and investigate Rabi oscillations in a periodically driven
three-level system under in- and off- resonance conditions. It was shown that
the Landau-Zener probabilities reveal two different patterns: "step" and
"beats"-like behaviours associated with the quantum interference effects.
Control on these two regimes is provided by change of the ratio between two
characteristic time scales of the problem. We demonstrate through the analysis
of a periodically driven three-level system, that if a direct transition
between certain pairs of levels is allowed and fine-tuned to a resonance, the
problem is mapped to the two-level Rabi model. If the transition between pair
of levels is forbidden, the off-resonance Rabi oscillations involving second
order in tunneling processes are predicted. This effect can be observed by
measuring a population difference slowly varying in time between the states of
the Cooper-pair box characterised by the same parity.
