Understanding two-photon double ionization of helium from the perspective of the characteristic time of dynamic transitions

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.