Two-electron processes can generate high harmonics beyond the conventional single-active-electron cutoff. Motivated by recent experimental evidence of an extended secondary plateau in the helium high-harmonic spectrum [S. Wang et al, Optica, (2023); S. Wang et al, In Print in Nature Photon., (2026)], we present a two-electron generalisation of the strong-field approximation. We analyse the resulting expressions using the saddle-point method and determine the extended cutoff. We find good agreement with classical predictions of cutoff scalings of $4.7$ and $5.5$ times the ponderomotive energy, which significantly exceed the established single-electron scaling of 3.17. We calculate high-harmonic spectra generated via a two-electron process in helium atoms driven by an intense few-cycle infrared laser pulse. Our results demonstrate that the harmonic spectrum extends far beyond the water window, reaching photon energies up to $\approx 1.2\,\mathrm{keV}$ in the soft x-ray region. The large spectral bandwidth can support the generation of sub-attosecond soft x-ray pulses, which are of particular interest for probing ultrafast dynamics across matter, including applications in core-level spectroscopy and biological imaging.
Two-electron processes can generate high harmonics beyond the conventional single-active-electron cutoff. Motivated by recent experimental evidence of an extended secondary plateau in the helium high-harmonic spectrum [S. Wang et al, Optica, (2023); S. Wang et al, In Print in Nature Photon., (2026)], we present a two-electron generalisation of the strong-field approximation. We analyse the resulting expressions using the saddle-point method and determine the extended cutoff. We find good agreement with classical predictions of cutoff scalings of 4.7 and 5.5 times the ponderomotive energy, which significantly exceed the established single-electron scaling of 3.17. We calculate high-harmonic spectra generated via a two-electron process in helium atoms driven by an intense few-cycle infrared laser pulse. Our results demonstrate that the harmonic spectrum extends far beyond the water window, reaching photon energies up to ≈1.2keV in the soft x-ray region. The large spectral bandwidth can support the generation of sub-attosecond soft x-ray pulses, which are of particular interest for probing ultrafast dynamics across matter, including applications in core-level spectroscopy and biological imaging.
We show that a simple nonperturbative two-level model of an atom driven by a very strong periodic field, results in a rich picture of very high harmonic generation and related phenomena. It reproduces experimentally observed plateau, yields for the first time simple analytic formulas for the plateau cutoff frequency, critical driving intensity, and saturation, and predicts intensity-induced multi-resonances. One of the most fascinating phenomenon discovered recently in nonlinear interaction of light with atoms and ions, is very high-order (up to 135) odd harmonic generation (HHG) by intense (∼ 1013 W/cm2 and higher) optical laser radiation in rare gasses and some ions [1], The spectra of generated harmonics drasticly deviate from the perturbation theory predictions [2], In particular, intensity of harmonics, falling monotonically with their orders only up to a certain point, levels off forming a so-called “plateau", and falls monotonically again beyond it. Generally, harmonte generation depends on phase-matching conditions and nonlinear response of individual atoms. It has recently become clear however [3,4] that the major features of HHG, in particular the plateau, result mainly from general properties of atomic nonlinear response. The most direct and apparently successful way so far to approach the problem theoretically has been numerical simulation of the Schrödinger [2,5-7] (including an empiric rule [6]) for many-electron atoms using Hartry-Slater approximation. This approach requires, however, tremendous amount of calculations and involves many processes, making it difficult to gain simple insights. An interesting simplified model [8] based on 3-D delta-potential with a single (ground) level [9] produces results in integration form. The idea of retaining a single energy scale (ionization energy) brings one close to an even simpler system: a two-level model atom. A two-level model of HHG [10], however, due to various complications introduced into it, failed to generate simple results, whereas an analytic solution [11] holds for a virtually degenerate two-level model only (see below) which is unapplicable to the experimental conditions [1-4,7,9]; besides, no relaxation was considered in [10,11].
Carrier-envelope-phase-dependent effects of high-harmonic generation for two-level atoms interacting with Gaussian shape few-cycle pulses (with $\ensuremath{\phi}=0$ and $\ensuremath{\phi}=\ensuremath{\pi}/2$) have been investigated theoretically by using a time-frequency analysis. The differences in energies and populations of the two adiabatic states and the harmonic spectra in the cutoff region are illustrated. We find that, by tuning the carrier-envelope phase, the high-harmonic generation in the cutoff region can be controlled. Our results also show that the model of two-level atoms driven by few-cycle pulses can be used to study carrier-envelope phase effects.
High-order harmonic spectra and attosecond pulse generation from Rydberg atom (He[Formula: see text]) driven by the spatially inhomogeneous field have been theoretically investigated. (i) Firstly, with an electron initially in a single excited Rydberg state (nth), the harmonic yield can be enhanced due to the decreased ionization potential, and a maximum enhancement can be obtained when the initial state is prepared as the third excited state (n = 3). However, the low cutoff energy from the excited state is unbeneficial to the generation of the higher photon pulse. Thus, with the further introduction of the laser chirp, not only the harmonic cutoff is extended, but also the harmonic modulation is reduced. As a result, five super-bandwidths from 63 eV to 267 eV can be found. (ii) Secondly, by preparing the initial state as a coherent superposition of excited state, the harmonic yield can be further enhanced, especially for the coherent superposition of the first and the third (n = 1 + 3) and the second and the fourth (n = 2 + 4) excited states, the harmonic yield is enhanced by 4–8 orders of magnitude compared with the case of the single ground initial state. Furthermore, by properly adding the laser pulse into the spatially inhomogeneous region (gap center [Formula: see text] a.u.) from left [Formula: see text] to right [Formula: see text], much higher cutoff energies can be obtained in the left region. As a consequence, two super-bandwidths of 248 eV and 496 eV can be obtained. Finally, by properly superposing the harmonics, a series of sub-25-as pulses with intensity enhancement of 5–8 orders of magnitude can be produced.
Extending the high-harmonic cutoff with experimentally accessible fields is essential for advancing tabletop coherent extreme ultraviolet (EUV) and soft X-ray sources. Although terahertz (THz) assistance offers a promising route, cutoff extension at weak, laboratory-accessible THz strengths remain poorly understood. In this report, we comprehensively investigate THz-assisted high-order harmonic generation (HHG) using time-dependent Schrödinger equation simulations supported by classical trajectory analysis and Bohmian-based quantum dynamics. By mapping the plateau evolution versus THz strength, we show that even weak THz fields can extend the cutoff, producing a pronounced ``fish-fin'' structure whose prominent rays saturate near Ip+8Up. We trace this extension to long electron excursions spanning several optical cycles before recombination, and provide a fully consistent explanation using both classical analysis and Bohmian trajectories flow. Our findings reveal that this cutoff-extension mechanism is remarkably robust, persisting across different atomic species and remaining insensitive to variations in the driving parameters. These results demonstrate that cutoff control is achievable with laboratory-scale THz fields, offering practical guidelines for engineering coherent high-energy HHG, and providing a robust pathway for tracking ultrafast electron motion in real time.
The past thirty years witnessed rapid developing of strong field physics that leads to the first generation of attosecond (10−18 s) pulses. The duration of available pulses has recently been reduced to less than 100 as, approaching the Kepler period of a classical electron revolving around the nucleus. For the first time, electron motions inside atoms, molecules and solids can be traced, probed and even steered in real time. It pushed the study on ultrafast dynamic processes into a new frontier, giving birth to attosecond physics. The development of attosecond pulses provides unprecedented coherent ultrashort XUV (extreme ultra-violet) of soft X-ray sources that enable imaging the atomic structure and resolving the related quantum dynamics for matter science. In particular, it allows the probing and monitoring of electron dynamics in the natural time scale of attoseconds with atomic spatial resolution. The generation and applications of attosecond pulses are directly related to the control and manipulation of the sub-cycle electron dynamics. Recent studies on the correlated multielectron dynamics have provided insight into the optical and dynamical properties of matter in the perspectives of time, phase and entanglement. It deepens our understanding on some of the fundamental questions, e.g., how the single or multiple photons are absorbed and how short are the quantum processes such as photoionization, tunneling ionization and charge migration. Although currently attosecond pulses are limited by the flux and pulse duration, the combination of attosecond pulses with intense infrared laser pulses paves new ways of controlling and probing electron dynamics in the domains of energy, time and space that lays the foundation for petahertz optoelectronics and attosecond pulse based transient spectroscopy. In this review, we will first briefly summarize the advancement of strong field and ultrafast physics including the relevant experimental findings, the theoretical explorations, the related technologies and the possible applications. Then we focus on the sub-cycle electron or multielectron dynamics and their manifestation on the ionization and coherent radiation properties. We start with the time behavior of Fano resonance and how it can be probed in time domain. In section 2, the correlated electron dynamics is discussed in term of anti-screening effect and Pauli effect that influence strong field ionization and high harmonic generation processes. In section 3, the coherent emission with frequency from terahertz to soft XUV and their joint measurement are discussed by emphasizing the generation mechanism of terahertz wave in two-color laser pulses. The developed high-harmonic and terahertz wave spectroscopy (HATS) is shown capable of imaging the molecular structure and dynamics. In section 4, the time-resolving of electron and hole dynamics and their coherence is proposed based on attosecond transient absorption technique. The coherence of ionization and the correlated electron dynamics is hinted crucial for electron-hole interaction. The topics are very limited and mainly selected from the perspectives of our group.
Caustics and catastrophes in strong-field physics − Picard-Lefschetz theory as a universal approach to saddle-point methods in attosecond science
Ultrashort laser pulses on the attosecond timescale are typically achieved via high-order harmonic generation (HHG), a nonlinear process in which atoms interact with intense light fields to emit a broad spectrum of harmonics. HHG is commonly described in terms of a `quantum orbits' model based on several interfering electron trajectories, thereby incorporating both quantum-mechanical effects and an intuitive picture of classical dynamics. By tuning the parameters of the driving laser field, the interplay between these trajectories can be controlled, shaping the emitted light. Mathematically, this model expresses the harmonic response as a highly oscillatory integral. Applying saddle-point methods to this integral allows it to be decomposed into contributions from distinct saddle points of the semi-classical action, thereby linking quantum dynamics to classical trajectories. However, a general framework for applying these methods across arbitrary parameters and laser configurations has been missing.
In this thesis, we introduce Picard-Lefschetz theory and develop practical numerical methods for its application. These enable the evaluation of oscillatory integrals and identification of contributions from individual critical points.
We apply these techniques to strong-field ionisation and HHG, focusing on caustics − enhancement features where trajectories coalesce and standard approximations fail. Our methods remain valid in these regions, allowing systematic analysis of parameter regimes and revealing previously inaccessible features. This work improves the understanding and control of ultrafast light-matter interactions.