  A parametrically modulated oscillator has two opposite-phase vibrational
states at half the modulation frequency. An extra force at the vibration
frequency breaks the symmetry of the states. The effect can be extremely strong
due to the interplay between the force and the quantum fluctuations resulting
from the coupling of the oscillator to a thermal bath. The force changes the
rates of the fluctuation-induced walk over the quantum states of the
oscillator. If the number of the states is large, the effect accumulates to an
exponentially large factor in the rate of switching between the vibrational
states. We find the factor and analyze it in the limiting cases. We show that
in the zero-temperature limit the extra force breaks the detailed balance,
leading to a nonperturbatively strong increase of the switching rate.
