Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2608.13686

Strong-field Herman-Kluk propagator method for high-harmonic generation in molecules

Phi-Hung Tran, Hao Quan Truong, R. Esteban Goetz, Anh-Thu Le

physics.atom-phquant-phSubmitted Aug 13, 2026• Announced Aug 17, 2026
arXivHTMLPDF

Abstract

We extend our recently developed semiclassical strong-field Herman-Kluk (SFHK) propagator method to calculate high-order harmonic generation (HHG) in diatomic molecules driven by few-cycle intense laser fields. On the example of applications to H2 and N2, we show that our method, based on a combination of the Herman-Kluk propagator and the strong-field approximation, can provide very accurate results for both HHG yield and phase, nearly identical to those from the exact numerical solutions of the time-dependent Schrodinger equation. To compare with experimental measurements, averaging over molecular orientations must be performed. Here we demonstrate a distinct and powerful advantage of the SFHK, as its Monte Carlo sampling for the integration over the alignment distribution can be efficiently combined with the integration over the initial momentum distributions of electron wave-packet right after the tunnel exit. Therefore, the total number of trajectories used for the alignment-averaged HHG spectrum does not increase much compared to that for a single fixed alignment. Similar to atomic targets, the main computational task in the SFHK is to solve the classical Hamiltonian equations for the active electron in the combined electron-target ion potential and electron-laser interaction. The motion of the center of each electron wave packet in the continuum, represented by a coherent state, is governed by an independent classical trajectory so that the computation can be parallelized very efficiently.

Comments
12 pages, 9 figures

Related papers

Highly accurate semiclassical strong-field Herman-Kluk propagator method for high-harmonic generation

Phi-Hung Tran, Hao Quan Truong, R. Esteban Goetz +1 more

We extend our recently developed semiclassical strong-field Herman-Kluk propagator (SFHK) method to calculate high-order harmonic generation (HHG) for atoms in intense lasers. We show that our method, based on a combination of the Herman-Kluk propagator and the strong-field approximation, can provide highly accurate results for both HHG yield and phase, nearly identical to those from the exact numerical solutions of the time-dependent Schrödinger equation. We provide detailed analyses of our method and its applications to the HHG process, particularly the recombination time. The main computational task in this method is to solve the classical Newton equations for the active electron in the combined atomic potential and laser-electron interaction. The motion of the centers of the electron wave packets, modeled by coherent states, is governed by independent classical trajectories so that the computation can therefore be parallelized very efficiently.

physics.atom-phCross-listedarXiv:2601.17164•Jan 27, 2026

Comment on Semiclassical approximations in phase space with coherent states

Frank Großmann, Michael F. Herman

The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.

Journal of Physics A Mathematical and General•Oct 21, 2002•38 citations

Quantum optical analysis of high-order harmonic generation in H$_2^+$ molecular ions

Javier Rivera-Dean, Philipp Stammer, Andrew Maxwell +5 more

We present a comprehensive theoretical investigation of high-order harmonic generation in H$_2^+$ molecular ions within a quantum optical framework. Our study focuses on characterizing various quantum optical and quantum information measures, highlighting the versatility of HHG in two-center molecules towards quantum technology applications. We demonstrate the emergence of entanglement between electron and light states after the laser-matter interaction. We also identify the possibility of obtaining non-classical states of light in targeted frequency modes by conditioning on specific electronic quantum states, which turn out to be crucial in the generation of highly non-classical entangled states between distinct sets of harmonic modes. Our findings open up avenues for studying strong-laser field-driven interactions in molecular systems, and suggest their applicability to quantum technology applications.

arXiv:2307.12381•Jul 23, 2023•0 citations

High-harmonic generation driven by temporal-mode quantum states of light

Juan M. González-Monge, Felipe Reibnitz Willemann, Johannes Feist

We develop a theoretical framework for high-harmonic generation (HHG) driven by quantum states of light based on a temporal-mode expansion of the electromagnetic field. This approach extends previous single plane-wave mode treatments to realistic pulse configurations and arbitrary multi-mode states of light, resolving conceptual inconsistencies arising from non-normalizable infinite plane waves and establishing consistency between analytical and numerical methods. We derive a correction factor that quantifies deviations from the diagonal approximation (in which the yield becomes a statistical average over classical-field simulations) both for the response of a single atom and in the many-atom regime. Our results confirms that the HHG spectrum for atoms driven by any quantum state of light in free space is accurately described by averaging semi-classical calculations over the Husimi distribution, with no observable genuine quantum effects in the spectrum. We also demonstrate that in the many-atom regime, the mean-field coherent-state approximation underlying this treatment does not preserve probabilities, although unitarity is restored by in the diagonal approximation. The absence of genuine quantum effects in the HHG yield is attributed to the large photon numbers ($\sim 10^{11}$) required to reach HHG intensities in free space, which render quantum fluctuations negligible. We discuss nanophotonic environments with ultrasmall mode volumes as potential platforms where few-photon strong-field processes could exhibit genuine quantum signatures.

quant-pharXiv:2512.06602•Dec 9, 2025•0 citations

About the role of short and long trajectories on the quantum optical state after high-harmonic generation

Javier Rivera-Dean

High-harmonic generation (HHG) involves the up-conversion of a high-intensity driving field into its harmonic orders. This process is intrinsically non-classical, requiring from quantum mechanics for a complete explanation as, under suitable conditions, involves phenomena such as particle tunneling through a potential barrier. When exposed to a high-intensity, low-frequency laser field, bound electrons ionize via tunneling, accelerate under the driving field, and recombine with the parent ion, emitting high-harmonic radiation. However, electrons can follow two distinct pathways -- short and long trajectories -- during these steps. In this work, we evaluate the signatures left by these trajectories on the quantum optical state after HHG, and observe that they lead to entanglement between the driving field and the generated harmonics. By leveraging these correlations, we use harmonic generation to herald the creation of optical Schrödinger cat-like states in the driving field. Additionally, using an ab-initio approach, we examine how propagation effects, which spatially separate the harmonic contributions from short and long trajectories, influence the non-classical characteristics of the emitted light.

arXiv:2409.15556•Sep 23, 2024•0 citations

Efficient Algorithms for Semiclassical Quantum Dynamics

David Sattlegger

We study the well-known Herman-Kluk propagator in order to calculate approximate solutions to the time--dependent semiclassical Schroedinger equation in high dimensions. We derive a discretisation scheme and provide its approximation properties. This scheme, as well as other semiclassical methods, are of an intrinsically parallel nature. This allows for the design of highly efficient algorithms employing state of the art parallelization and vectorization techniques.

•Jan 1, 2016•0 citations
We extend our recently developed semiclassical strong-field Herman-Kluk (SFHK) propagator method to calculate high-order harmonic generation (HHG) in diatomic molecules driven by few-cycle intense laser fields. On the example of applications to H2 and N2, we show that our method, based on a combination of the Herman-Kluk propagator and the strong-field approximation, can provide very accurate results for both HHG yield and phase, nearly identical to those from the exact numerical solutions of the time-dependent Schrodinger equation. To compare with experimental measurements, averaging over molecular orientations must be performed. Here we demonstrate a distinct and powerful advantage of the SFHK, as its Monte Carlo sampling for the integration over the alignment distribution can be efficiently combined with the integration over the initial momentum distributions of electron wave-packet right after the tunnel exit. Therefore, the total number of trajectories used for the alignment-averaged HHG spectrum does not increase much compared to that for a single fixed alignment. Similar to atomic targets, the main computational task in the SFHK is to solve the classical Hamiltonian equations for the active electron in the combined electron-target ion potential and electron-laser interaction. The motion of the center of each electron wave packet in the continuum, represented by a coherent state, is governed by an independent classical trajectory so that the computation can be parallelized very efficiently.
We extend our recently developed semiclassical strong-field Herman-Kluk propagator (SFHK) method to calculate high-order harmonic generation (HHG) for atoms in intense lasers. We show that our method, based on a combination of the Herman-Kluk propagator and the strong-field approximation, can provide highly accurate results for both HHG yield and phase, nearly identical to those from the exact numerical solutions of the time-dependent Schrödinger equation. We provide detailed analyses of our method and its applications to the HHG process, particularly the recombination time. The main computational task in this method is to solve the classical Newton equations for the active electron in the combined atomic potential and laser-electron interaction. The motion of the centers of the electron wave packets, modeled by coherent states, is governed by independent classical trajectories so that the computation can therefore be parallelized very efficiently.
The semiclassical Herman–Kluk initial value approximation of the quantum-mechanical propagator, which is heavily used in dynamical studies of atomic and molecular systems, takes into account nonlinearity of the underlying classical dynamics by using multiple classical initial value solutions. In applications to the propagation of Gaussian wavepackets non-Gaussian distortions in the course of time and therefore a realistic description of the quantum dynamics are possible. This is in contrast to the single-trajectory approximations investigated in a recent paper by Baranger et al , one of which was erroneously termed Herman–Kluk approximation.
Quantum optical analysis of high-order harmonic generation in H2+_2^+2+​ molecular ions
We present a comprehensive theoretical investigation of high-order harmonic generation in H2+_2^+2+​ molecular ions within a quantum optical framework. Our study focuses on characterizing various quantum optical and quantum information measures, highlighting the versatility of HHG in two-center molecules towards quantum technology applications. We demonstrate the emergence of entanglement between electron and light states after the laser-matter interaction. We also identify the possibility of obtaining non-classical states of light in targeted frequency modes by conditioning on specific electronic quantum states, which turn out to be crucial in the generation of highly non-classical entangled states between distinct sets of harmonic modes. Our findings open up avenues for studying strong-laser field-driven interactions in molecular systems, and suggest their applicability to quantum technology applications.
We develop a theoretical framework for high-harmonic generation (HHG) driven by quantum states of light based on a temporal-mode expansion of the electromagnetic field. This approach extends previous single plane-wave mode treatments to realistic pulse configurations and arbitrary multi-mode states of light, resolving conceptual inconsistencies arising from non-normalizable infinite plane waves and establishing consistency between analytical and numerical methods. We derive a correction factor that quantifies deviations from the diagonal approximation (in which the yield becomes a statistical average over classical-field simulations) both for the response of a single atom and in the many-atom regime. Our results confirms that the HHG spectrum for atoms driven by any quantum state of light in free space is accurately described by averaging semi-classical calculations over the Husimi distribution, with no observable genuine quantum effects in the spectrum. We also demonstrate that in the many-atom regime, the mean-field coherent-state approximation underlying this treatment does not preserve probabilities, although unitarity is restored by in the diagonal approximation. The absence of genuine quantum effects in the HHG yield is attributed to the large photon numbers (∼1011\sim 10^{11}∼1011) required to reach HHG intensities in free space, which render quantum fluctuations negligible. We discuss nanophotonic environments with ultrasmall mode volumes as potential platforms where few-photon strong-field processes could exhibit genuine quantum signatures.
High-harmonic generation (HHG) involves the up-conversion of a high-intensity driving field into its harmonic orders. This process is intrinsically non-classical, requiring from quantum mechanics for a complete explanation as, under suitable conditions, involves phenomena such as particle tunneling through a potential barrier. When exposed to a high-intensity, low-frequency laser field, bound electrons ionize via tunneling, accelerate under the driving field, and recombine with the parent ion, emitting high-harmonic radiation. However, electrons can follow two distinct pathways -- short and long trajectories -- during these steps. In this work, we evaluate the signatures left by these trajectories on the quantum optical state after HHG, and observe that they lead to entanglement between the driving field and the generated harmonics. By leveraging these correlations, we use harmonic generation to herald the creation of optical Schrödinger cat-like states in the driving field. Additionally, using an ab-initio approach, we examine how propagation effects, which spatially separate the harmonic contributions from short and long trajectories, influence the non-classical characteristics of the emitted light.