  We show that the noise spectrum of a parametrically excited nonlinear
oscillator can display a fine structure. It emerges from the interplay of the
nonequidistance of the oscillator quasienergy levels and quantum heating that
accompanies relaxation. The heating leads to a finite-width distribution over
the quasienergy, or Floquet states even for zero temperature of the thermal
reservoir coupled to the oscillator. The fine structure is due to transitions
from different quasienergy levels, and thus it provides a sensitive tool for
studying the distribution. For larger damping, where the fine structure is
smeared out, quantum heating can be detected from the characteristic
double-peak structure of the spectrum, which results from transitions
accompanied by the increase or decrease of the quasienergy.
