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Journal of the Optical Society of America B

Solvable approximate model for the harmonic radiation from atoms subjected to oscillatory electric fields

Lawrence C. Biedenharn, George A. Rinker, J. C. Solem

Published Feb 1, 1989• 14 citations
Publisher

Abstract

We develop a nonperturbative model for the response of atomic electrons to oscillating external electric fields, valid in the physical regime where the central electric force is well described by an inverse square law and where the driven oscillations preserve the principal quantum number. We use the model to calculate the odd-harmonic electric-dipole radiation from a single atom driven by a laser field. The exact time-dependent eigensolutions of the model illustrate Zel’dovich’s concept of quasi-energy for sinusoidally time-dependent Hamiltonians. The validity and applicability of the model and suggested improvements are briefly discussed.

DOI
10.1364/josab.6.000221

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We develop a nonperturbative model for the response of atomic electrons to oscillating external electric fields, valid in the physical regime where the central electric force is well described by an inverse square law and where the driven oscillations preserve the principal quantum number. We use the model to calculate the odd-harmonic electric-dipole radiation from a single atom driven by a laser field. The exact time-dependent eigensolutions of the model illustrate Zel’dovich’s concept of quasi-energy for sinusoidally time-dependent Hamiltonians. The validity and applicability of the model and suggested improvements are briefly discussed.
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In quantum theory, the description of radiation by a system of charged particles in stationary and nonstationary cases requires different approaches. While radiative processes in stationary systems are covered in textbooks, the emission of radiation by nonstationary systems driven by an external time-dependent force has not received due attention. In this paper, we develop a quantum theory describing the spectrum of photons emitted by a generic nonstationary atomic system and show how to implement it in practical calculations. We also discuss the approximation in which the exact quantum formula for the spectrum reduces to the well-known classical formula with the classical dipole moment replaced by the expectation value of the dipole moment calculated quantum mechanically. This approximation underlies the classical ansatz commonly used in the theory of high-order harmonic generation (HHG). To illustrate the theory, we apply it to calculating HHG spectrum for a one-dimensional zero-range-potential model. The spectra obtained from the quantum formula and from the ansatz are different and the difference grows as the probability of ionization grows. This difference is an observable effect predicted by the present theory.
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The subject of this chapter is the quantum mechanical analysis of the interaction of electromagnetic radiation with atomic transitions. The analysis is based on the Schrödinger wave equation, and in the first section, the gauge-invariant form of the external electromagnetic field is introduced. The electric dipole interaction and the long-wavelength approximation for the analysis of this interaction are discussed. The perturbative analysis of both single-photon and two-photon electric dipole interactions is presented, and density matrix analysis is introduced. The interaction of radiation with the resonances of atomic hydrogen is then discussed. The analysis is performed for both coupled and uncoupled representations. In the last section of the chapter, the radiative interactions for multielectron atoms are discussed. The Wigner–Eckart theorem and selection rules for transitions between levels characterized by coupling are developed. The effect of hyperfine splitting on radiative transitions is also briefly discussed.
An external electric field is provided to the familiar one-dimensional quantum harmonic oscillator model with a various position mass m(x)=(a m_0)/(a+x). The Nikiforov-Uvarov approach is involved to investigate a particular solution to the exact Schrödinger equation. The exactly solvable confined model of the quantum harmonic oscillator in an external electric field was proposed. Starting with the BenDanial-Duke kinetic energy operator approximations, the construction of the position-dependent mass Schrödinger equation is studied. The analytic representation of the wave functions of the stationary states is expressed analytically and graphically using modified Laguerre polynomials as well as the energy spectrum. In contrast with the absence of an external electric field, when the energy spectrum totally overlaps with that of the harmonic oscillator potential in an external electric field: the energy spectrum becomes non-equidistant and varies depending on some factors. The Nikiforov-Uvarov approach succeeds broadly in demonstrating the wave function and energy spectrum and showing good sense.