TDPT

calculation of the two-photon double ionization of helium with the TDPT2 (second order time dependent perturbation theory)

$$ P(E_1, E_2) = 1/2(\frac{c}{4\pi^2})^2|\sqrt{\frac{\sigma^{He}(E_1)\sigma^{He^+}(E_2)}{\omega_{ai}\omega_{fa}}}K(E_a) + \sqrt{\frac{\sigma^{He}(E_{2})\sigma^{He^+}(E_1)}{\omega_{ai}\omega_{fa}}}K(E_b)|^2 $$

where $K(E_a)=\int_{-\infty}^{\infty}\int_{-\infty}^{\tau_1}F(\tau_1)F(\tau_2)e^{i((Ea-Ei)\tau_1+(Ef-Ea)\tau_2)}d\tau_1d\tau_2$

reference

Jiang, W.-C., Xiong, W.-H., Zhu, T.-S., Peng, L.-Y. & Gong, Q. Double ionization of He by time-delayed attosecond pulses. J. Phys. B At. Mol. Opt. Phys. 47, 091001 (2014).