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Quadrupole Photoionization Time Delay For Ne And Ar

Pratikshya Parajuli

Published Jul 24, 2026• 0 citations

Abstract

Early investigations of the photoionization time delay focused solely on dipole contributions at low-energy. Later experimental progress demonstrated that non-dipole time delays are also measurable. This motivated us to initiate a theoretical program aimed at studying detailed quadrupole time delays in the noble gasses Ne and Ar and assessing their physical significance. In this work, we examined the angular dependence of the quadrupole photoemission time delay for Ne and Ar using the Relativistic-Random-Phase Approximation (RRPA) method. Our approach began with identifying the relevant transitions for each sub-shell and obtaining the corresponding expression for the transition amplitude. From these amplitudes, we extracted the phase and computed the associated non-dipole time delay as a function of emission angle and energy for each sub-shell. Where applicable, we evaluated the time delay across both low- and high-energy regions and compared subshells that share the same principal quantum number n. These comparisons reveal the substantial influence of relativistic interactions between different subshells and atomic species. Currently, we have obtained results for Ne and Ar for the subs-hells ns1/2, np1/2, np3/2. Future work will extend these calculations to additional atomic systems, including heavier noble gasses, various atomic ions, and confined atomic environments.

DOI: 10.57709/367

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Early investigations of the photoionization time delay focused solely on dipole contributions at low-energy. Later experimental progress demonstrated that non-dipole time delays are also measurable. This motivated us to initiate a theoretical program aimed at studying detailed quadrupole time delays in the noble gasses Ne and Ar and assessing their physical significance. In this work, we examined the angular dependence of the quadrupole photoemission time delay for Ne and Ar using the Relativistic-Random-Phase Approximation (RRPA) method. Our approach began with identifying the relevant transitions for each sub-shell and obtaining the corresponding expression for the transition amplitude. From these amplitudes, we extracted the phase and computed the associated non-dipole time delay as a function of emission angle and energy for each sub-shell. Where applicable, we evaluated the time delay across both low- and high-energy regions and compared subshells that share the same principal quantum number n. These comparisons reveal the substantial influence of relativistic interactions between different subshells and atomic species. Currently, we have obtained results for Ne and Ar for the subs-hells ns1/2, np1/2, np3/2. Future work will extend these calculations to additional atomic systems, including heavier noble gasses, various atomic ions, and confined atomic environments.

Anisotropy parameters describing the angular dependence of the photoionization delay are defined. The formalism is applied to results obtained with the relativistic random phase approximation with exchange for photoionization delay from the outermost

s

-orbitals in selected rare-gas atoms. Any angular dependence in the Wigner delay is induced here by relativistic effects, while the measurable atomic delay exhibits such a dependence also in the nonrelativistic limit. The contributions to the anisotropy from the different sources are disentangled and discussed. For the heavier rare gases, it is shown that measurements of the delay for electrons ejected in specific angles, relative to, e.g., those ejected along the laser polarization, are directly related here to the Wigner delay. For a considerable range of angles, the contributions from the second photon largely get canceled when the results in different angles are compared, and this angle-relative atomic delay is then close to the corresponding Wigner delay.

We solve the time-dependent Schr\"odinger equation for a noble gas atom (He and Ne) driven by an ionizing XUV and dressing IR fields. From this solution we deduce an angular dependence of the photoemission time delay as measured by the RABBITT (reconstruction of attosecond beating by interference of two-photon transitions) technique. We use a recent angle-resolved RABBITT measurement on helium [S. Heuser et al., Phys. Rev. A 94, 063409 (2016)] to test and calibrate our theoretical model. Based on this calibration, we find no significant difference between the time delay in He measured in the angle-integrated RABBITT experiments [C. Palatchi et al., J. Phys. B 47, 245003 (2014) and D. Gu\'enot et al., J. Phys. B 47, 245602 (2014)] and measured or calculated in the polarization axis direction. The angular dependence of the photoemission time delay of Ne is shown to be qualitatively different from He because of the different orbital character of the valence

2p

orbital. The angular momentum projection dependence of the time delay in Ne is also investigated.

Formulas to calculate the atomic photoionization delay have been derived taking into account the interaction between the channels and the scattering of a photoelectron. The photoionization delay difference between Ne 2p and 2s shells has been calculated for photon energies up to 200 eV. The calculated delay difference at a photon energy of 105 eV is 18.5 as, which is approximately 8 as larger than delay differences previously calculated taking into account only the interactions between the channels and is in agreement within the experimental error with two existing experimental values.

We present first-principles numerical simulations of photoionization in neon induced by bichromatic extreme ultraviolet pulses with frequencies

ω

and

2ω

, specially chosen to make

ω

equal to the energy difference between the

2s

and

2p

subshells. This allows for the production of photoelectrons from the

2s

shell by

2ω

pulse and from the

2p

shell by

ω

pulse with the same energy. Using the multi-configurational time-dependent Hartree-Fock method, we explore how Rabi coupling between subshells generates coherence between the corresponding photoelectron wave packets. Our \textit{ab initio} calculations confirm the analytical results derived from the essential-states approach in [K. L. Ishikawa, K. C. Prince, and K. Ueda, J. Phys. Chem. A 127, 10638 (2023)], validating the theoretical predictions. Although we focus on the Ne

2p

and

2s

subshells, our approach is applicable to a broad range of systems exhibiting photoionization from multiple subshells. The laser parameters employed in our simulations are available in modern Free Electron Lasers (FELs), and we anticipate that this work could stimulate experimental investigations using FELs to study ion-photoelectron coherence and entanglement.

An atom confined in an optical dipole trap is a promising candidate for a qubit. Analyzing the temporal response of such trapped atoms enables us to estimate the speed at which quantum computers operate. The present work models an atom in an optical dipole trap formed using crossed laser beams and further examines the photoionization time delay from such confined atoms. We study noble gas atoms, such as Ne (Z = 10), Ar (Z = 18), Kr (Z = 36), and Xe (Z = 54). The atoms are considered to be confined in an optical dipole trap using X-ray Free Electron Lasers (XFEL). The present work shows that the photoionization time delay of the trapped atoms is different compared with that of the free atoms. This analysis alerts us that while talking about the speed of quantum computing, the temporal response of the atoms in the trapped environment must also be accounted for.