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

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.