The generations of the high-order harmonic and the attosecond pulse from He+ ion driven by the chirped pulse combined with the unipolar pulse have been numerically investigated. The results show that by preparing the initial state as the coherent superposition of the ground and the first excited states of He+ ion, the harmonic yield is enhanced by 6 orders of magnitude in comparison with the single ground initial state case. Further, by introducing the laser chirp and the unipolar controlling pulse, the cutoff energy of harmonic spectrum is remarkably enhanced, and the interference structure of harmonic is also reduced. Further, with the introduction of the spatial inhomogeneous effect, the harmonic cutoff can be further extended, resulting in a super-bandwidth of 320 eV. By superposition of harmonics, three attosecond X-ray pulses with durations from 45 as to 48 as can be obtained, which is 5 ~ 6 orders of magnitude improvement in comparison with the single ground initial state case.
