  By using the B-spline numerical method, we investigate a two-photon
double-ionization (TPDI) process of helium in a high-frequency laser field with
its frequency ranging from 1.6~a.u. to 3.0~a.u. and the pulse duration ranging
from 75 to 160~attoseconds. We found that there exists a characteristic time
$t_{c}$ for a TPDI process, such that the pattern of energy distribution of two
ionized electrons presents a peak or two, depending respectively on whether the
pulse duration is shorter or longer than $t_{c}$. Especially, as the pulse
duration is larger than $t_c$, the TPDI spectrum shows a double-peak structure
which is attributed to the fact that most of the electron-electron Coulomb
interaction energy is acquired by single electron during their oscillation
around the nucleus before the two electrons leave. Additionally, if the photon
energy is less than the ionization energy of He$^{+}$, $t_{c}$ is not a fixed
value, and it increases as the photon energy decreases; while if the energy of
a photon is greater than the ionization energy of He$^{+}$, $t_{c}$ is fixed at
about 105 attoseconds. We further found that, for a helium-like ion in its
ground state, the characteristic time for the case of the photon energy larger
than the ionization energy of the second electron has a key relation with the
Coulomb interaction energy $\overline{V}_{12}$ between the two electrons, which
can be expressed as $t_{c}\overline{V}_{12}=4.192$, a type of quantum
mechanical uncertainty relation between time and energy. In addition, this
relation can be attributed to the existence of a minimal evolution time from
the ground state to a double ionization state with two electrons carrying
different energies. These results may shed light on deeper understanding of
many-electron quantum dynamical processes.
