Time-dependent perturbation theory beyond the dipole approximation for two-photon ionization of atoms

We develop a time-dependent perturbation theory (PT) beyond the dipole approximation to analyze the two-photon ionization dynamics of atoms exposed to short laser pulses with an arbitrary polarization. In a wide range of laser parameters, the good performance of the PT method is validated by comparing the results of a number of physical quantities with those calculated by numerically solving the time-dependent Schrödinger equation beyond the dipole approximation. Subsequent applications of the PT method in the nonresonant regime allow us to unveil the important role of the interferences between the dipole and nondipole transition pathways to the photoelectron momentum shift along the laser-propagation direction. In particular, we find that the ratio of the probability of each dipole transition channel to the total ionization probability oscillates with the increase of the photon energy. The interferences among different pathways and the oscillatory behavior of the ratios jointly lead to a series of minima in the linear momentum transfer for the case of the linear polarization.

Figure 1

(a) Photoelectron energy distribution $P\left(E\right)$ and (b) angular distribution $\mathrm{\Delta }{P}_{nd}\left(\theta \right)$ from the hydrogen ground state exposed to a linearly polarized ${cos}^2$-shaped pulse with ${\omega }_0=0.28$ a.u. at various pulse durations: $N=5$ (the red open circles for the PT), $N=10$ (the blue triangles for PT), $N=25$ (the black open squares for the PT), $N=100$ (the purple solid circles for the PT). In both (a) and (b), solid lines represent the TDSE results. In (a), results for different pulse durations are normalized for a better visibility. In (b), the results for $N=5, N=10$, and $N=25$ have been multiplied by a factor 3 for clarity.

Figure 2

(a) Photoelectron energy distribution $P\left(E\right)$ for the hydrogen $1s$ state exposed to a linearly polarized ${cos}^2$-shaped pulse with $N=5$ at three different ${\omega }_0$: 0.29 (the green dash-dotted line for PT), 0.375 a.u. (the light blue dashed line for PT), and 0.48 a.u. (the orange dotted line for PT). The solid curves are from the TDSE. (b) Total ionization probabilities at various ${\omega }_0$ for different pulse durations: $N=5$ (the red open circles for PT), $N=25$ (the black open squares for PT), and $N=100$ (the purple solid circles for PT). Solid lines represent the TDSE results.

Figure 3

Expectation of the linear momentum transferred to the photoelectron ionized from (a) the hydrogen and (b) the helium atomic system. The two-photon ionization probability is shown as a function of the central frequency ${\omega }_0$ for (c) the hydrogen and (d) the helium atomic systems. The square symbols in (a) and (c) are results from the PT method.

Figure 4

(a) Photoelectron momentum shift $〈p_z〉$ varies with the photon energy of a pulse with the linear polarization (LP) or the right-handed circular polarization (RCP). The red solid (blue dashed) lines are for the TDSE results of the LP (RCP). The red open squares (blue open circles) represent $〈p_z〉$ of the LP (RCP) obtained by Eq. (20) in the PT method. (b) ${〈p_z〉}_{1,2}$ of the LP from PT. (c) Purple solid (black dashed) line indicates the TDSE results of $P_1/P_t$ ($P_2/P_t$) with the symbols from the PT results.