  Vacuum radiation has been the subject of theoretical study in both cosmology
and condensed matter physics for many decades. Recently there has been
impressive progress in experimental realizations as well. Here we study vacuum
radiation when a field mode is driven both parametrically and by a classical
source. We find that in the Heisenberg picture the field operators of the mode
undergo a Bogolyubov transformation combined with a displacement, in the
Schr\"odinger picture the oscillator evolves from the vacuum to a squeezed
coherent state. Whereas the Bogolyubov transformation is the same as would be
obtained if only the parametric drive were applied the displacement is
determined by both the parametric drive and the force. If the force is applied
well after the parametric drive then the displacement is the same as would be
obtained by the action of the force alone and it is essentially independent of
$t_f$, the time lag between the application of the force and the parametric
drive. If the force is applied well before the parametric drive the
displacement is found to oscillate as a function of $t_f$. This behavior can be
understood in terms of quantum interference. A rich variety of behavior is
observed for intermediate values of $t_f$. The oscillations can turn off
smoothly or grow dramatically and decrease depending on strength of the
parametric drive and force and the durations for which they are applied. The
displacement depends only on the Fourier component of the force at a single
resonant frequency when the forcing and the parametric drive are well separated
in time. However for a weak parametric drive that is applied at the same time
as the force we show that the displacement responds to a broad range of
frequencies of the force spectrum. Implications of our findings for experiments
are briefly discussed.
