  Parametric amplification of attosecond coherent pulses around 100 eV at the
single-atom level is demonstrated for the first time by using the 3D
time-dependent Schr{\"o}dinger equation in high-harmonic generation processes
from excited states of He$^+$. We present the attosecond dynamics of the
amplification process far from the ionization threshold and resolve the physics
behind it. The amplification of a particular central photon energy requires the
seed XUV pulses to be perfectly synchronized in time with the driving laser
field for stimulated recombination to the He$^+$ ground state and is only
produced in a few specific laser cycles in agreement with the experimental
measurements. Our simulations show that the amplified photon energy region can
be controlled by varying the peak intensity of the laser field. Our results
pave the way to the realization of compact attosecond pulse intense XUV lasers
with broad applications.
