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Journal of Physics B Atomic Molecular and Optical Physics

Harmonic generation with ionizing two-level atoms

Enrique Conejero Jarque, Luis Plaja

Published Apr 28, 1998• 16 citations
Publisher

Abstract

We study the harmonics generated by a two-level system in which we include the effects of ionization and recombination. The ionized wavepackets are treated as free classical particles moving inside the electric field. We recover the basic features of the single-atom harmonic generation in the tunnelling limit, finding a good agreement between our model and the one-dimensional Schrödinger equation results. The model allows us to investigate the role of bound states in the harmonic generation in the tunnelling regime. In particular, we show that the harmonic efficiency in the plateau region is strongly dependent on the strength of the dipole moment of bound-bound transitions.

DOI
10.1088/0953-4075/31/8/018

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Physical Review A•Apr 1, 1996•134 citations
We study the harmonics generated by a two-level system in which we include the effects of ionization and recombination. The ionized wavepackets are treated as free classical particles moving inside the electric field. We recover the basic features of the single-atom harmonic generation in the tunnelling limit, finding a good agreement between our model and the one-dimensional Schrödinger equation results. The model allows us to investigate the role of bound states in the harmonic generation in the tunnelling regime. In particular, we show that the harmonic efficiency in the plateau region is strongly dependent on the strength of the dipole moment of bound-bound transitions.
We present a study of the effect of time-dependent ionization on harmonic generation by bound–bound transitions. The introduction of tunneling and above-barrier ionization rates in a simple bound system, a two-level atom, induces an unexpected richness in the harmonic spectra, leading to the production of radiation at high frequencies. Such frequencies are not reached when no ionization or steady ionization is considered. Time-dependent ionization is, therefore, a key process in the understanding of generation of middle plateau frequency harmonics from atoms in the tunneling ionization regime and shows that bound–free transitions are not uniquely responsible for the higher-harmonic frequencies in realistic atoms. Following previous authors’ studies, we use an adiabatic approach to understand the presence of these higher frequencies.
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.
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].
We investigate the third order harmonic generation for model atoms interacting with intense short laser pulses. We focus on the cases where the ponderomotive energy is not big enough to generate high-order harmonics, and where resonant transitions between bound states are also as important as multi-photon ionization and low-order harmonic generation. We find that the third order harmonic intensity from a model hydrogen atom can decrease as the laser frequency draws near a resonance between bound states. We explain this result by looking into the ionization and Rabi oscillation from the numerical calculation of the time-dependent 1-D Schrödinger equation. In order to support our explanation we also investigate the populations of bound states that play an important role in this atomic process.
A simple, "exactly solvable" atomic model will be presented that helps to discuss some of the interesting features of multiphoton ionization of atoms. The electromagnetic field is treated non-perturbatively, it can be strong. The atomic electrons are assumed to be non-interacting (Hartree model). They are bound in a harmonic oscillator potential (one- or three-dimensional). At the ionization energy, the potential is cut off and the free electron has no Coulomb attraction. The Schrodinger equation has exact analytic solutions valid in the inner or outer regions. If the field is turned on adiabatically, the exact solution can be found by matching these solutions at the boundary; both of their value and the normal derivative at all times. Thus the machine calculation reduces to repeated application of fast Fourier transforms and to the solution of linear equations (unfortunately a lot of them).
In this Rapid Communication we present a method of extending the harmonic generation plateau to high energies with very high conversion efficiencies using moderate intensities. Preparing the initial state as a coherent superposition of the ground state and an excited state we can induce dipole transitions between the continuum and the ground state via the intermediate (excited) state responsible for the ionization. We solve numerically the Schr\"odinger equation and show that the spectrum contains two distinct plateaus, with cutoffs at Ue+3Up{U}_{e}+3{U}_{p}Ue​+3Up​ and Ug+3Up{U}_{g}+3{U}_{p}Ug​+3Up​. Using an essential states method we demonstrate that the plateaus are due to recombination into the excited and ground states. We briefly discuss possible experimental realizations of the scheme we present.