It is known that the dynamics of two (Coulomb-interacting) nonrelativistic electrons confined by a parabolic potential and driven by a classical, intense laser field (in dipole approximation) is exactly soluble. We calculate the time-dependent population of the harmonic oscillator states and the energy absorbed from the laser. It turns out that the key entity on which all observables sensitively depend is the modulus square of the Fourier-transformed vector potential of the laser field, evaluated at the harmonic oscillator frequency. The system is transparent to laser field configurations for which this entity vanishes. We discuss the Poisson statistics behaviour of the transition probabilities and analyse the conditions for the complete survival and full depletion of the initial state.