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

Quantum Electronics and Laser Science Conference

Analytic plateau-cutoff formula for high harmonic generation in two-level atoms

A. E. Kaplan, P. L. Shkolnikov

Published May 2, 1993• 0 citations
OpenAlex

Abstract

On the basis of our new theory of very-high-harmonic generation (HHG) in a superdriven two-level atom (see our paper for poster session), we have derived, for the first time to our knowledge (considering here all the existing models,1 including a numerical simulation of a two-level model driven by short pulses2), an amazingly simple analytic formula for one of the most important characteristics of the phenomenon: the size of the so-called plateau, or its cutoff frequency ωcut.

Related papers

Superdressed two-level atom: Very high harmonic generation and multiresonances

A. E. Kaplan, P. L. Shkolnikov

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 the experimentally observed plateau, yields simple analytic formulas for the plateau cutoff frequency, critical driving intensity, and saturation, and predicts intensity-induced multiresonances.

Physical Review A•Feb 1, 1994•66 citations

Super-Dressed Two-Level Atom: Very High Harmonic Generation and Multi-Resonances

A. E. Kaplan, P. L. Shkolnikov

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].

Shortwavelength: Physics with Intense Laser Pulses•Jan 1, 1993•1 citation

Modeling harmonic generation by a degenerate two-level atom

R. Burlon, G. Ferrante, C. Leone +2 more

An analytical theory of the generation of high-order harmonics of laser radiation has been developed on the basis of a two-level model atom with degenerate levels. Among other parameters, onset, width, and cutoff of the plateau in the harmonic spectrum are obtained in simple analytical forms that connect the basic problem parameters and permit a transparent interpretation of the mechanism underlying the spectrum formation for this specific case. Selected numerical calculations are reported to corroborate the analytical findings and to investigate other harmonic-spectrum features.

Journal of the Optical Society of America B•Jan 1, 1996•7 citations

Super-dressed two-level atom: Very high harmonic generation and related phenomena

A. E. Kaplan, P. L. Shkolnikov

We present here a two-level model that not only reproduces experimentally observed plateau, but also allows us to evaluate its characteristics in explicit analytic form and predict new dramatic effects. Rare gases and rare-gas like ions in which high order odd harmonic generation (HHG) has been observed so far, may not be inconsistent with two-level description as far as HHG is concerned.>

IEEE Lasers and Electro-Optics Society Annual Meeting (LEOS)•Dec 30, 2002•0 citations

Superdriven two-level atom: generation of very-high-order harmonics and combination frequencies

A. E. Kaplan, P. L. Shkolnikov

Very high harmonic generation (HHG) by intense optical laser in rare gasses has recently been observed [1] in experiment.

Annual Meeting Optical Society of America•Jan 1, 1993•0 citations

Phase of harmonics from strongly driven two-level atoms

F. I. Gauthey, Christoph H. Keitel, P. L. Knight +1 more

Harmonic generation is studied in an idealized model medium in which the laser field essentially interacts only with a single atomic transition and where ionization can be neglected. Such two-level models are known to generate high harmonics with a plateau and a cutoff. We put forward a quantitative rule for the cutoff frequency in the harmonic spectrum in this model which we relate to the ``${\mathrm{I}}_{\mathrm{P}}$+${3\mathrm{U}}_{\mathrm{P}}$ rule'' in real systems with ionization. We examine the phase properties of the emitted harmonics and show that their coherence and thus phase matching is more favorable in the cutoff regime than in the plateau of the harmonic spectrum. This behavior is known to occur in realistic models of neon, and that this occurs also for the simple idealized two-level model strongly suggests that the sudden change of phase coherence from the cutoff to the plateau regime is as intrinsic a property of nonlinear systems as are the plateau and the cutoff.

Physical Review A•Jan 1, 1997•41 citations
On the basis of our new theory of very-high-harmonic generation (HHG) in a superdriven two-level atom (see our paper for poster session), we have derived, for the first time to our knowledge (considering here all the existing models,1 including a numerical simulation of a two-level model driven by short pulses2), an amazingly simple analytic formula for one of the most important characteristics of the phenomenon: the size of the so-called plateau, or its cutoff frequency ωcut.
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].
An analytical theory of the generation of high-order harmonics of laser radiation has been developed on the basis of a two-level model atom with degenerate levels. Among other parameters, onset, width, and cutoff of the plateau in the harmonic spectrum are obtained in simple analytical forms that connect the basic problem parameters and permit a transparent interpretation of the mechanism underlying the spectrum formation for this specific case. Selected numerical calculations are reported to corroborate the analytical findings and to investigate other harmonic-spectrum features.
Harmonic generation is studied in an idealized model medium in which the laser field essentially interacts only with a single atomic transition and where ionization can be neglected. Such two-level models are known to generate high harmonics with a plateau and a cutoff. We put forward a quantitative rule for the cutoff frequency in the harmonic spectrum in this model which we relate to the ``IP{\mathrm{I}}_{\mathrm{P}}IP​+3UP{3\mathrm{U}}_{\mathrm{P}}3UP​ rule'' in real systems with ionization. We examine the phase properties of the emitted harmonics and show that their coherence and thus phase matching is more favorable in the cutoff regime than in the plateau of the harmonic spectrum. This behavior is known to occur in realistic models of neon, and that this occurs also for the simple idealized two-level model strongly suggests that the sudden change of phase coherence from the cutoff to the plateau regime is as intrinsic a property of nonlinear systems as are the plateau and the cutoff.