The interaction between strong laser fields and matter has emerged as a prominent tool for probing the internal structure of atoms and molecules and field-induced ultrafast electron dynamics. During the multiphoton ionization of atoms and molecules by intense laser pulses, ionized electron wave packets from different paths interfere, resulting in complex interference patterns in the photoelectron momentum distributions (PMDs). Over the past decades, a prominent interference structure known as strong-field photoelectron holography (SFPH) has been observed. In molecule fields, researchers use holographic structures to probe molecular structure and orientation dynamics information, but no relevant literature has been found in the atomic field. By numerically simulating the interaction between the excited state 2pz of a hydrogen atom and linearly polarized laser pulses with different polarization directions, we can extract the structural information of atomic orbitals from the PMDs. In addition, we also discuss a feasible pump-probe scheme for experimental validation.

To simulate atomic ionization in a linearly polarized laser field, we numerically solve the three-dimensional time-dependent Schrödinger equation (TDSE) in the velocity gauge with dipole approximation. We use the finite-element discrete variable representation (FE-DVR) method to discretize the radial part of the wave function. For the time evolution of the wave function, we use the split-Lanczos method. After the laser pulse concludes, the ionization probability is extracted from the final wave function by projecting it onto the scattering state.

The configuration of the present laser-atom interaction is illustrated in Fig. 1. The quantization axis of the state 2pz is along the z-axis. Two polarization directions of the laser pulse, Θ=0 [Fig. 1(a)] and π/6 [Fig. 1(b)], are presented. The wavelength, pulse duration, and peak intensity of the laser pulse are fixed to be 2000 nm, 10 optical cycles, and 1013 W/cm², respectively. The PMDs at different angles Θ are given in Fig. 2. Different angles indeed give rise to different PMDs. We can observe the PMDs are symmetrical with respect to the laser polarization at Θ=0 and π/2 [Figs. 2(a) and 2(d)], while such symmetry is broken at Θ=π/6 and π/3 [Figs. 2(b) and 2(c)]. In the tunneling ionization regime, the symmetry of the distribution of the initial transverse momentum of electrons depends on the Fourier transform of the initial wave function. Based on adiabatic approximation theory, we found that the symmetry of both holographic and fan-shaped interference structures closely depends on the initial transverse momentum distribution of the direct electrons. Next, we investigate how tunneling filters with spherically symmetric and non-spherically symmetric orbits affect the initial transverse momentum distribution of electrons (Fig. 3). For the 2pz orbital, the transverse momentum k⊥|ψ2p is symmetric only when k‖=0 and is asymmetric for other values [Fig. 3(d)]. Clearly, the asymmetrical PMDs exactly mimic the asymmetrical momentum distribution of the initial orbital. To quantitatively study the correlation between the initial orbital and the PMDs, we define a parameter ΔY to describe the asymmetry. The research found that the asymmetry of the initial orbital, denoted as ΔY2p, qualitatively describes the changing trend of the ionized electron distribution ΔY with Θ increasing (Fig. 4). Therefore, the asymmetry parameter of the final electron reflects the information of atomic orbital structure. We extend our discussion to the multi-photon ionization and transition ionization regime in Fig. 5(a), the asymmetry parameter ΔY2p still well reproduces the Θ-dependence of the photoelectron asymmetry ΔY after extending the ionization from the tunneling to the multi-photon and transition regime. Therefore, we can generally conclude that the asymmetry in photoelectron distribution correlates with the asymmetry of the initial-state momentum distribution. We show the dependence of the asymmetry parameter ∆Y on the Keldysh parameter γat a specific angle Θ=π/4 in Fig. 5(b). In the tunneling regime γ<1, the asymmetry parameter ∆Y is around 0.3 with slight fluctuations. However, the fluctuations become significant in the transition and multi-photon ionization regime γ>1. This is because in the transition and multiphoton ionization regions, there are multiple resonant ionization channels, making it difficult to maintain consistency between the PMDs and the initial transverse momentum distribution. Experimental verification of the present theoretical predictions requires a pump-probe scheme, as the excited state 2pz is not naturally largely populated. We should use a pump laser pulse to prepare the excited state 2pz before it interacts with the probe pulse. The configuration of the pump and probe laser pulses is illustrated in Fig. 6(a). The PMDs in the pump-probe scheme are shown in Fig. 6(b). We observe that the result is highly consistent with that in Fig. 2(c). To better understand the potential impact of the pump-probe method on extracting ionization electron asymmetry, we further investigated the influence of pump duration and the time delay between the two laser pulses on the extraction of asymmetry parameters in Figs. 6(c) and 6(d). We present a theoretical approach to probe atomic orbital structure information and investigate the correlation between atomic orbits and final state momentum distributions under different ionization mechanisms. Finally, we consider implementing feasible pump-probe detection schemes to validate its predictions.

We have theoretically investigated the photoionization of the excited state 2pz of hydrogen atoms by linearly polarized laser pulses. We identified asymmetrical PMDs with respect to the laser polarization direction. In the tunneling ionization regime, this asymmetry arises from the asymmetrical distribution of the initial orbital with respect to the polarization direction, resulting in an unequal transverse momentum distribution of the initial electrons. In both tunneling and multi-photon ionization regimes, the asymmetry parameter ∆Y of the PMDs as a function of the laser polarization direction Θ is qualitatively reproduced by the asymmetry parameter ΔY2p of the initial orbital. Our theoretical prediction could be experimentally verified in a pump-probe scheme. Our calculation indicates that the asymmetry parameter ∆Y of the PMDs can be well extracted even if the population of the excited state 2pz after the pump pulse ends is not large.