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Parametric amplification of attosecond coherent pulses around 100 eV at thesingle-atom level is demonstrated for the first time by using the 3Dtime-dependent Schr{\"o}dinger equation in high-harmonic generation processesfrom excited states of He$^+$. We present the attosecond dynamics of theamplification process far from the ionization threshold and resolve the physicsbehind it. The amplification of a particular central photon energy requires theseed XUV pulses to be perfectly synchronized in time with the driving laserfield for stimulated recombination to the He$^+$ ground state and is onlyproduced in a few specific laser cycles in agreement with the experimentalmeasurements. Our simulations show that the amplified photon energy region canbe controlled by varying the peak intensity of the laser field. Our resultspave the way to the realization of compact attosecond pulse intense XUV laserswith broad applications.

Parametric attosecond pulse amplification far from the ionization threshold from high order harmonic generation in He$^+$