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Journal of the Optical Society of America B

Symmetries in harmonic generation by a two-color field: an application to a simple model atom

Saverio Bivona, R. Burlon, C. Leone

Published Jun 1, 1999• 12 citations
Publisher

Abstract

We show that the parity of the order of the harmonics generated by a bichromatic field with commensurate frequencies may be related to the invariance properties of the time-dependent Hamiltonian with respect to some suitable transformations. To simplify the mathematical treatment, we have used a simple model atom to perform calculations that show the effects of the relative phase of the driving fields. Harmonic generation has been described as a two-step process that involves replicas of the Stark levels. In agreement with other previous treatments in which other model atoms were used, we have found that a small contamination of the fundamental harmonic can produce significant modifications of the emission spectra.

DOI
10.1364/josab.16.000986

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We show that the parity of the order of the harmonics generated by a bichromatic field with commensurate frequencies may be related to the invariance properties of the time-dependent Hamiltonian with respect to some suitable transformations. To simplify the mathematical treatment, we have used a simple model atom to perform calculations that show the effects of the relative phase of the driving fields. Harmonic generation has been described as a two-step process that involves replicas of the Stark levels. In agreement with other previous treatments in which other model atoms were used, we have found that a small contamination of the fundamental harmonic can produce significant modifications of the emission spectra.
The generation of harmonics by atoms or ions in a two-color, coplanar field configuration with commensurate frequencies is investigated through both an analytical calculation based on the Lewenstein model and the numerical ab initio solution of the time-dependent Schr\"odinger equation of a two-dimensional model ion. Through the analytical model, selection rules for the harmonic orders in this field configuration, a generalized cutoff for the harmonic spectra, and an integral expression for the harmonic dipole strength are provided. The numerical results are employed to test the predictions of the analytical model. The scaling of the cutoff as a function of both one of the laser intensities and frequency ratio $\ensuremath{\eta}$ as well as entire spectra for different $\ensuremath{\eta}$ and laser intensities are presented and analyzed. The theoretical cutoff is found to be an upper limit for the numerical results. Other discrepancies between analytical model and numerical results are clarified by taking into account the probabilities of the absorption processes involved.
Abstract Linearity is a fundamental postulate of quantum mechanics which is occasionally the subject of debate. This paper investigates the possibility of checking this assumption by using a laser field. We study the corrections caused by the presence of a small nonlinearity in the Hamiltonian of a quantum system. As a model we use a simplified two-level quantum system whose states are coupled by a small off-diagonal term proportional to the population of the upper level. The nonlinearity causes spontaneous decay of the upper level, shift and broadening of the line and the sensitive dependence of the final state on the initial condition. The presence of a strong laser field, resonant with the atomic transition, enhances the population transfer among the levels and introduces quantitative and qualitative modifications of the spectra of high order harmonic generation (HHG); these are cumulative effects which can be subject to experimental checks. Experiments are needed in order to set an upper limit to the nonlinear term.
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].
Abstract The contribution of excited discrete atomic levels to the generation of high- and low-order harmonics in the interaction of a single atom with a two-color laser field has been studied for the first time. The studies were carried out for two types of atoms with different symmetry of the ground state wave functions (Ag, Ne) and for a two-color laser field consisting of the linearly polarized fundamental and second harmonics of Ti:Sa laser ( ω + 2 ω ). The angle between the polarization of the ω + 2 ω laser field components is varied. A significant effect of the symmetry of populated excited levels on the polarization characteristics of the generated harmonics is shown. A new method of control over the polarization state of the generated radiation is proposed.
We present a theoretical study of the main characteristics of two-color photoionization spectra obtained when atoms are subjected to an intense radiation pulse containing the fundamental of an infrared laser and one of its higher harmonics. The frequency of the latter is chosen to be high enough that the atom can be ionized by a single photon. The calculation is performed by numerical integration of the time-dependent Schrödinger equation for a (three-dimensional) hydrogen atom. We present the general properties of the spectra and angular distributions as well as the possible occurrence of interferences between the laser-assisted process and above-threshold ionization. Our results indicate that such interference effects, which can lead to partial coherent control of the photoionization process, can be observed at moderate laser intensities within the reach of currently operated laser and harmonic sources.
Phase-dependent resonance transitions induced in two-level atoms by a bichromatic field are reported. Strong phase effects are observed in the interaction of a fast atomic beam with a standing-wave rf field. In the atomic rest frame, this field consists of two components with different frequencies and a controlled initial phase difference. Numerical calculations including well-defined phase factors are performed in the time-dependent density matrix formalism. The observed effects are in excellent agreement with this theory.