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Optics Express

Probing nonperturbative characteristics in the retrieval of resonant photoionization time delay by a bicircular two-color interferometry

Xiaoyun Zhao, Mingqing Liu, Yanjun Chen

Published Jan 30, 2026• 1 citation
Publisher

Abstract

We theoretically propose a bicircular two-color interferometry to retrieve the resonant photoionization time delay (RPTD) of atomic hydrogen, which can be described by the energy derivative of amplitude phase difference between single- and two-photon ionization pathways. Our results reveal that the RPTD can be well retrieved from the maximal asymmetry of photoelectron momentum distributions in the perturbative and nonperturbative regimes. In the perturbative regime, the RPTD can be characterized by perturbation theory, which demonstrates a decreasing trend with the increasing energy in counter-rotating pulse configurations, while in the co-rotating case, which exhibits pronounced oscillatory characteristics. These oscillations originate from quantum interference between photoelectron wave packets generated through direct single-photon ionization processes and those mediated by resonant excitation pathways caused by the broad spectral bandwidth of ultrashort laser pulses. In the nonperturbative regime, we develop the strong-field nonperturbation theory (SFNPT) by taking the population depletion and Stark energy shift of the ground state into account, which shows a good agreement with the numerical solution of the time-dependent Schrödinger equation. The RPTD shows a smoothly increasing characteristic as photoelectron energy increases for the counter-rotating and co-rotating cases, which differs remarkably from the case in the perturbative regime. Our SFNPT model unveils that this abnormal phenomenon can be attributed to the population depletion of the ground state. Our proposed bicircular two-color interferometry approach provides an avenue for exploring the resonant time delays in atomic photoionization and observing their nonperturbative characteristics.

DOI
10.1364/oe.586534

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We theoretically propose a bicircular two-color interferometry to retrieve the resonant photoionization time delay (RPTD) of atomic hydrogen, which can be described by the energy derivative of amplitude phase difference between single- and two-photon ionization pathways. Our results reveal that the RPTD can be well retrieved from the maximal asymmetry of photoelectron momentum distributions in the perturbative and nonperturbative regimes. In the perturbative regime, the RPTD can be characterized by perturbation theory, which demonstrates a decreasing trend with the increasing energy in counter-rotating pulse configurations, while in the co-rotating case, which exhibits pronounced oscillatory characteristics. These oscillations originate from quantum interference between photoelectron wave packets generated through direct single-photon ionization processes and those mediated by resonant excitation pathways caused by the broad spectral bandwidth of ultrashort laser pulses. In the nonperturbative regime, we develop the strong-field nonperturbation theory (SFNPT) by taking the population depletion and Stark energy shift of the ground state into account, which shows a good agreement with the numerical solution of the time-dependent Schrödinger equation. The RPTD shows a smoothly increasing characteristic as photoelectron energy increases for the counter-rotating and co-rotating cases, which differs remarkably from the case in the perturbative regime. Our SFNPT model unveils that this abnormal phenomenon can be attributed to the population depletion of the ground state. Our proposed bicircular two-color interferometry approach provides an avenue for exploring the resonant time delays in atomic photoionization and observing their nonperturbative characteristics.
Abstract Atomic photoionization time delay is a vital component of attosecond metrology for investigating the ultrafast electron dynamics in atomic systems on their natural time scale. Here, we probe the factor which leads to the nonperturbative characteristics of resonant photoionization time delay (RPTD) of atomic hydrogen by numerically solving the time-dependent Schrödinger equation (TDSE) and the strong-field nonperturbation theory (SFNPT) with considering the population depletion and ac-Stark energy shift of ground state. RPTD can be retrieved from the energy-resolved optical phase of a bicircular two-color field corresponding to the maximal asymmetry of photoelectron angular distribution even in the nonperturbative regime. Our results reveal that as laser intensity increases the RPTD shows a remarkable difference of energy-dependent behaviors. At lower intensity, the absolute value of RPTD displays a smooth drop with increasing electron energy, however, at higher intensity its absolute value exhibits an obvious enhancement. Given the agreement between TDSE and SFNPT, we clearly elucidate the physical origin of nonperturbative characteristics, i.e., the population depletion of ground state plays a crucial role rather than ac-Stark energy shift. More specifically, the population depletion severely affects the Fourier transformation of the laser envelope, which is an important phase component of transition amplitude. Our work clarifies the physical mechanism of atomic photoionization time delay in nonperturbative environments, advancing the comprehensive understanding of time in attosecond science.
We propose a protocol to retrieve the photoionization time delay that employs a pair of oppositely circularly polarized attosecond extreme ultraviolet pulses with frequencies $\ensuremath{\omega}$ and $2\ensuremath{\omega}$. By numerically solving the three-dimensional time-dependent Schr\"odinger equation, we observe an oscillation in the energy-resolved photoionization time delay encoded in the relative optical phase between $\ensuremath{\omega}$ and $2\ensuremath{\omega}$ pulses when the emission asymmetry of the photoelectron is maximal. Our results show that this oscillation arises from the interference between two one-photon ionization pathways from the ground state due to the broad bandwidth effect of an ultrashort pulse: the absorption of one left-handed circularly polarized photon and the emission of one right-handed circularly polarized photon. The latter is regarded as a consequence of the breakdown of the rotating-wave approximation (RWA). We compare the perturbation theory with and without RWA and demonstrate that photon emission plays a crucial role in retrieving photoionization time delay. Our study enriches the underlying physics of attosecond spectroscopy and provides a framework for accurately measuring photoionization time delays in atoms and molecules.
When an atom is driven by a laser field with a photon energy that is nearly resonant with an atomic transition, it can undergo periodic population transfers between the ground and excited state, known as Rabi oscillations [1], [2]. The aim of this work is to study photoionization from atoms for parameters accessible to a free-electron laser (FEL) [3]. Using time dependent perturbation theory to model photoionization from a Rabi oscillating atom, we have shown that in addition to the expected pathway of ionization from the excited state, non-resonant two-photon ionization will have a comparable amplitude at intensities on the order of 1013W/cm2[3].
We study time delays in atomic multiphoton ionization (MPI) using the phase-sensitive detection of photoelectron vortices. Photoelectron vortices are created by MPI with counterrotating circularly polarized (CRCP) ultrashort double-pulse sequences. The vortices are characterized by a spiral-shaped photoelectron momentum distribution (PMD) resulting from the Ramsey-type interference of the partial photoelectron wave packets created by each pulse. The slope of the spiral arms encodes the time delay between the partial wave packets with interferometric precision. We study the ionization time delay in the 1+21+21+2 resonance-enhanced multiphoton ionization of potassium atoms using femtosecond CRCP white-light supercontinuum pulses. Pairwise interference of the partial wave packets with a reference wave packet gives rise to vortices of different rotational symmetry, which are superimposed to form the total PMD. The PMD is reconstructed tomographically and decomposed into the vortices using three-dimensional Fourier analysis. From the energy-dependent spectral phases of the vortices, we retrieve a resonant ionization delay on the order of 1 fs, much shorter than the pulse duration. Our results demonstrate the power of photoelectron vortices for the accurate determination of photoionization time delays and open up new perspectives for photoelectron chronoscopy including the measurement of ionization delays on the attosecond timescale.
We investigate resonant strong field ionization of atomic hydrogen with respect to the 1s–2p transition. By 'strong' we understand that Rabi periods are executed on a femtosecond time scale. Ionization and AC Stark shifts modify the bound state dynamics severely, leading to nonperturbative signatures in the photoelectron spectra. We introduce an analytical model, capable of predicting qualitative features in the photoelectron spectra such as the positions of the Autler–Townes peaks for modest field strengths. Ab initio solutions of the time-dependent Schrödinger equation show a pronounced shift and broadening of the left Autler–Townes peak as the field strength is increased. The right peak remains rather narrow and shifts less. This result is analysed and explained with the help of exact AC Stark shifts and ionization rates obtained from Floquet theory. Finally, it is demonstrated that in the case of finite pulses as short as 20 fs the Autler–Townes duplet can still be resolved. The fourth generation light sources under construction worldwide will provide bright, coherent radiation with photon energies ranging from a tenth of an meV up to tens of keV, hence covering the regime studied in the paper so that measurements of nonperturbative, relative AC Stark shifts should become feasible with these new light sources.
We theoretically investigate the resonant two-photon ionization of a hydrogen atom in the counter-rotating circularly polarized double-pulse sequences, by solving the time-dependent Schr\"odinger equation. Due to the induced Rabi oscillations between the ground and excited states, the well-known Autler-Townes doublet is observed in the photoelectron spectra. More interestingly, our numerical results reveal an additional peak located between the Autler-Townes doublets, which originates from single-photon ionization from the excited state by the second pulse. By adjusting the laser intensity of the two pulses, the photoelectron yields of the three peaks can be sensitively modulated, under the control of completed Rabi cycles within the first pulse. In addition, our results show a potential of manipulating the photoemission anisotropy by both the laser intensity and the two-pulse time delay, through adjusting the competition of different partial waves generated by the two pulses.