Introduction
Multiphoton processes in the continuum are attracting increasing attention within the attosecond community. Recently, several theoretical frameworks have been developed to address this problem. Without loss of generality, these approaches can be categorized as either numerical or analytical methods. Building on the integration of established methodologies (Boll et al, 2025), we present analytical expressions for the matrix elements governing three-photon transitions, with applications to angularly resolved photoemission time delays. We benchmark our results against numerically exact solutions of the Time-Dependent Schrödinger Equation (TDSE) for hydrogen atoms.

Methods
To pursue our goal, we combine analytical and numerical methods to study angularly resolved time delays in simple atomic systems undergoing three-photon transitions.

Results and Discussion
Our findings show a transition from qualitative to quantitative agreement between analytic and exact numerical results for angularly resolved time delays at increasing photoelectron energy. Furthermore, differences observed at lower kinetic energies may be ascribed to the asymptotic description of intermediate states.

Conclusions
In summary, we demonstrate that analytic (radial) matrix elements, which convey information about the angular quantum numbers of final states, are accurate enough to theoretically describe angularly resolved time delays in simple atomic systems. This assertion is valid for photoelectron energies above ~7 eV, with the quality of the analytic results improving at higher energies.

Ionization of molecules by electron impact represents one of the most fundamental interactions in nature, whose interest is relevant to a wide range of applications. Kinematically complete (e,2e) experiments, in which the energies and momenta of all final-state particles are determined, provide the most detailed information on the ionization reaction through the triple differential cross-section (TDCS). Compared to the case of atomic targets, performing such measurements on molecules is made difficult because of the close spacing between electronic states and also because of the contributions of rotational and vibrational states. Moreover, TDCS measurements on an absolute scale are scarce. Theoretically, finding an accurate quantal description of multicenter continuum states is a formidable task which necessarily requires some approximations.
To this end, we have proposed [1] a model, named M3CWZ, in which all continuum particles are represented by Coulomb waves with spatially variable charges; the model accounts for exchange effects and post-collision interactions.
Recently [2], we used this M3CWZ model to examine the ionization of water and methane molecules. For low impact energy on water, the results of our calculations satisfactorily reproduce absolute experimental data at 65 eV. For methane, satisfactory agreement is found but only for outer-shell ionization. As with other models presented in the literature, such as M3DW and MCTDW, experimental–theoretical agreement is not uniform, with several features left unexplained. It turns out that our M3CWZ model globally manages—at a moderate computational cost—to capture multicenter distortion effects, providing results similar in quality to those of other more sophisticated (but more computationally expensive) models.

[1] A. Tamin et al 2024 J. Chem. Phys. 161 16430.
[2] A. Tamin, Phd thesis, Sétif, Algeria (2025)

Abstract: Quantum information dynamics play a crucial role in understanding how information behaves under external electromagnetic influences, particularly laser fields. This study focuses on the interaction of a hydrogen atom with a laser field to explore scattering behavior and information transfer mechanisms. To achieve this, a wave function representing quantum information in the laser field is formulated. Using this wave function, the scattering matrix and transition matrix are calculated based on the Kroll–Watson approximation. The transition matrix is further employed to determine the differential cross section, which provides insights into the scattering dynamics of quantum information. Computational analysis reveals that the presence of a laser field significantly affects quantum information dynamics for both spin-up and spin-down states. The scattering angle and Bessel function parameters also show a noticeable influence on the scattering behavior. Results indicate that the scattering dynamics of quantum spin exhibit variable behavior with changes in the incident energy of the quantum information. Understanding the effect of laser fields on atomic systems provides essential insights into how quantum information can gain or lose coherence as it passes near or through atomic structures. This research contributes to the broader understanding of quantum information transmission, highlighting potential losses when information propagates through various materials or spatial regions. The findings can support the development of more efficient quantum communication systems and enhance control over quantum coherence in laser-assisted environments.

Laser spectroscopy stabilizes laser frequency to atomic resonances for cooling, clocks, and interferometry. Modulation Transfer Spectroscopy (MTS), a variant of Saturated Absorption Spectroscopy (SAS), locks a laser by modulating a single beam for precise, stable frequency control.

We characterize MTS on the D₂ lines of ⁸⁵Rb and ⁸⁷Rb using an electro-optic modulator. First, we study power broadening by symmetrically increasing probe and pump intensities. Next, we keep the probe near saturation while increasing pump power to optimize the locking signal [1,2].

Using 20 MHz rather than the conventional 5 MHz places the system in a fast-modulation regime where atoms cannot follow the modulation adiabatically. The MTS signal is dominated by four-wave mixing between the carrier and well-separated sidebands [1,4], which reduces the zero-crossing slope (lower Hz/V sensitivity) and complicates the line shape. Fast modulation can nonetheless improve rejection of low-frequency technical noise, reduce sensitivity to slow system drift, and separate the desired signal from other modulations present in the setup [3,4].

The 20 MHz choice therefore trades slope for noise immunity. We propose controlled power broadening as a practical route to mitigate the complex line-shape effects encountered in fast-modulation MTS [1,3,4].

# References

1. D. J. McCarron, S. A. King, S. L. Cornish, Modulation transfer spectroscopy in atomic rubidium, Meas. Sci. Technol. 19, 105601 (2008).
2. H.-R. Noh et al., Modulation transfer spectroscopy for 87Rb atoms: theory and experiment, Opt. Express 19, 23444–23452 (2011).
3. T. Preuschoff, M. Schlosser, G. Birkl, Optimization strategies for modulation transfer spectroscopy applied to laser stabilization, Opt. Express 26, 24010–24019 (2018).
4. E. Jaatinen, Theoretical determination of maximum signal levels obtainable with modulation transfer spectroscopy, Opt. Commun. 120, 91–97 (1995).

Fundamental constants—such as the fine-structure constant α, the strong-interaction scale, and particle masses—may vary in an expanding Universe. A spatial variation could help explain apparent fine tuning: we inhabit a region where the values permit life. Hints from quasar absorption spectra suggest a gradient in α, but decisive confirmation requires laboratory tests. Atomic clocks provide such tests and, through their exquisite stability, enable sensitive searches for new physics.

Interactions between dark matter and ordinary matter can induce temporal variation of constants. For low-mass bosonic dark matter produced after the Big Bang, the field behaves classically, yielding first-order effects in the coupling—an enormous advantage over traditional second-order responses. Using clock comparisons, existing bounds on scalar dark-matter couplings to photons, electrons, quarks, and the Higgs can be tightened dramatically; our analyses improved previous limits by up to 15 orders of magnitude.

We assess several promising clock candidates with enhanced sensitivity to α variation while offering accessible cooling E1 lines and small systematic effects.

Highly charged-ion clocks offer reduced systematics due to their compact size, with α -variation and dark-matter responses enhanced by 1 - 2 orders of magnitude.

The isomeric 8.4 eV nuclear transition in ²²⁹Th, recently laser-excited by multiple groups, opens a path to a nuclear clock with accuracy potentially exceeding the best optical atomic clocks. Because the nucleus is well shielded from environmental perturbations, systematic shifts can be intrinsically small; however, the surrounding electrons strongly mediate excitation and decay via the electronic-bridge mechanism and can modify both transition frequency and lifetime by orders of magnitude. The ²²⁹Th transition is exceptionally sensitive to physics beyond the Standard Model, with four orders of magnitude enhancement.

High-precision measurements of the bound-electron g factor in hydrogen and other few-electron ions provide stringent tests of quantum electrodynamics in the presence of a magnetic field and enable independent determination of fundamental constants. Interpreting these experiments requires equally accurate theoretical calculations, particularly for the electron self-energy correction representing a dominant QED contribution.

In this work, we focus on the vertex self-energy diagram for the bound-electron g factor. The contribution of this diagram suffers from ultraviolet (UV) divergences. In order to separate them out, an expansion of electron propagators in terms of binding potential is applied. The UV-divergent term is calculated in momentum space after a renormalization. In principle, the remainder of the vertex contribution can be evaluated in coordinate space, but the slow convergence of partial-wave expansions in coordinate space significantly limits the accuracy of calculations. In [1], it was proposed to additionally separate the next-to-divergent term of the potential expansion, the so-called one-potential contribution, and to treat it in momentum space. This considerably improved the convergence. However, the obtained closed-form expression for the one-potential contribution in momentum space was rather complicated. Based on a fruitful idea first proposed in [2], we have derived several approximations for the one-potential contribution. Separation of these approximations from the UV-finite term keeps the same effect on the convergence of the partial-wave expansions, but they can be calculated more easily.

1. V.A. Yerokhin, et al., Phys. Rev. A.69, 052503 (2004).
2. J. Sapirstein and K. T. Cheng, Phys. Rev. A. 108, 042804 (2023).

Methods of bound-state QED that treat the self-energy contributions to the g
factor of highly charged ions within the partial-wave expansion usually face the problem
of slow convergence of the latter. This work proposes a method aimed at accelerating
this convergence. Namely, we consider the vertex diagram contributing to the self-energy
correction to the bound-electron g factor. This diagram has an ultraviolet divergence. An
expansion of the electron propagators in terms of the binding potential V is used for its
analysis. The individual terms of this expansion are treated in coordinate and momentum
spaces [1]. There is no simple closed-form expression for the two-potential contribution
corresponding to the second power of V. The main idea of our method is to approximate
this contribution based on expressions for zero- or one-potential terms, which can be
calculated with high accuracy. This idea was first proposed for the self-energy correction
to energy levels [2]. Similar methods have also been successfully applied to the calculations of the two-electron [3] and two-loop [4, 5] self-energy contributions. A general analytical
derivation of momentum and angular integrals is presented, which makes it possible to
evaluate the discussed contribution for an arbitrary state of H-like ions.
1. V. A. Yerokhin et al., Phys. Rev. A 69, 052503 (2004).
2. J. Sapirstein, K. T. Cheng, Phys. Rev. A 108, 042804 (2023).
3. A. V. Malyshev et al., Phys. Rev. A 109, 062802 (2024).
4. V. A. Yerokhin et al., Phys. Rev. A 133 251803 (2024).
5. V. A. Yerokhin et al., Phys. Rev. A 111, 042820 (2025).

The dominant quantum electrodynamics correction to the hyperfine splitting of energy levels in hydrogen-like ions comes from a set of self-energy (SE) diagrams. The nonperturbative in alpha*Z (alpha is the fine-structure constant and Z is the nuclear charge number) evaluation of the corresponding correction has a long history; see, e.g., Ref. [1] and references therein. To the best of our knowledge, all previous calculations were performed using the Feynman gauge only. Nevertheless, the advantages of applying the Coulomb gauge in the case of the Lamb-shift calculations are well known [2]. The present work has two primary goals. First, we want to numerically check the gauge invariance of the set of SE diagrams for the hyperfine splitting. Second, we aim to study the benefits of using the Coulomb gauge for hyperfine-splitting calculations.
The self-energy correction is conveniently divided into three parts [1]: irreducible, reducible, and vertex parts. The treatment of the irreducible part is reduced to an evaluation of a nondiagonal matrix element of the first-order self-energy operator. Therefore, its calculation in the Coulomb gauge is straightforward. The renormalization of the reducible and vertex parts is performed following the results presented in Refs. [3]. Currently, our work is focused on the initially ultraviolet finite many-potential contribution, which is considered in coordinate space within the partial-wave expansion approach. Once completed, conclusions will be drawn regarding improvements in accuracy due to the use of the Coulomb gauge.

[1] V. A. Yerokhin and U. D. Jentschura, PRA 81, 012502 (2010).
[2] D. Hedendahl, J. Holmberg, PRA 85, 012514 (2012); V. A. Yerokhin et al., PRA 111, 012802 (2025).
[3] G. S. Adkins, PRD 27, 1814 (1983); 34, 2489 (1986).

1. Introduction
The investigation of electronic reactions in collisions between ions and molecules
is relevant to many fields, including plasma physics, astrophysics, medical physics,
and radiobiology. In particular, in plasma-facing applications, beryllium and boron
have emerged as promising candidate materials for plasma—wall interfaces. In the
literature, numerous theoretical studies have computed total cross-sections (TCSs)
for collisions between highly charged bare ions and neutral atoms. The interaction between dressed projectiles and neutral atoms has been investigated by Das et
al. [1] using perturbative methods such as the boundary-corrected continuum intermediate state (BCCIS) approximation, as well as by non-perturbative approaches
(see Ref. [2] for a review).
2. Methods
The present work aims to investigate single-electron capture processes from multi-
electron atoms induced by collisions with multi-charged, dressed projectiles at
intermediate and high energies. This process is studied within the Continuum
Distorted Wave (CDW) formalism, extending the recent development of Quinto
et al. [3]—originally formulated for hydrogen—to multi-electron targets. In this
work, the interaction between the projectile and the active electron is described
by the analytic Green–Sellin–Zachor (GSZ) potential. The electrons bound to
the projectile are treated as frozen during the collision. The final states of the
projectile are described by excited atomic wave functions [4].
3. Results
The results in terms of total cross-sections are compared with both experimental
measurements and available theoretical data over the energy range from 10 keV/u
to 10 MeV/u.
References
[1] Das M. et al. 1998 Phys. Rev. A 57 3573
[2] Hill C. et al. 2023 Nucl. Fusion 63 125001
[3] Quinto M. A. et al. 2025 Atoms 13 84
[4] Novikov N. V. 2015 Wave Function Value Database

Introduction: Precision spectroscopy's advancing accuracy offers powerful opportunities to test Quantum Electrodynamics (QED) and search for physics beyond the Standard Model. As experimental techniques improve, measurements of atomic energy levels and bound-particle g-factors now challenge the accuracy of fundamental QED calculations. This growing precision creates a critical need to refine computational methods for key radiative corrections, particularly the bound-electron self-energy—a dominant contribution to the Lamb shift.

Methods: The modern approach for renormalization and calculation of the bound-electron self-energy diagram relies on a potential expansion, where the electron propagator is expanded perturbatively in a Dyson series of nuclear interaction. A key challenge is the slow convergence of the resulting infinite partial-wave expansion for the many-potential contribution, typically computed in coordinate space. The recently proposed Sapirstein–Cheng scheme accelerates this convergence by employing a difference method, calculating an auxiliary term in both momentum and coordinate space, which yields a faster-converging series for extrapolation.

Results: In this work, we present a further optimization of this method. Based on a detailed analysis of the asymptotic behavior of the partial-wave series, we introduce a new parameter into the subtraction term. Since the asymptotic behavior of both the many-potential term and the difference series decays as 1/k², this parameter can be tuned to cancel the leading asymptotic term completely. This optimization enhances the efficiency of the numerical extrapolation, improving the final accuracy of the self-energy correction calculation by an additional one to two orders of magnitude.

Conclusions: By accelerating the convergence of the partial-wave expansion, the developed method significantly increases the precision of self-energy calculations for hydrogen-like systems. This advancement is crucial for matching the accuracy of ongoing high-precision spectroscopic experiments and provides a more robust computational framework for evaluating higher-order Feynman diagrams, thereby enabling more sensitive searches for New Physics beyond the Standard Model.