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Journal of Physics A Mathematical and General

Quantum electrodynamical theory for the natural shape of the spectral line

В. М. Шабаев

Published Dec 7, 1991• 20 citations
Publisher

Abstract

In order to study the natural spectral line shape in the framework of quantum electrodynamics the resonance photon scattering on the relativistic few-electron atom is considered. The general case of the few overlapping levels is studied. In the resonance approximation the calculation formulae for the differential and total scattering cross-sections are obtained. The equations allowing improvement of the resonance approximation are also derived.

DOI
10.1088/0305-4470/24/23/028

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In order to study the natural spectral line shape in the framework of quantum electrodynamics the resonance photon scattering on the relativistic few-electron atom is considered. The general case of the few overlapping levels is studied. In the resonance approximation the calculation formulae for the differential and total scattering cross-sections are obtained. The equations allowing improvement of the resonance approximation are also derived.
A variational Hamiltonian method for quantum field theory is applied to the problem of natural line shape. Within scalar quantum electrodynamics, coupled variational equations are derived for the scattering of photons from a scalar (pionic) atom. It is shown that the coupled equations have a continuous spectrum for all energies above the atomic ground state. Excited states of the atom appear as resonances in particular partial-wave cross sections. The line center and shape are derived from the behavior of the photon-atom scattering phase shifts.
Abstract An examination is made of the Green dyadic for the propagation functions of non-relativistic quantum electrodynamics in the Coulomb gauge div A = 0. The existence of false precursors is shown to be associated with singularities at the origin in certain momentum space representations of the propagators and this allows a prescription to be given to eliminate such precursors. This prescription is proved equivalent to an exact consideration of the near fields associated with quantum transitions. The analogous treatment for the state vectors in decaying systems is carried through and is applied in detail to the problems of the shape of natural spectral lines and the transition rate for stimulated emission. It is shown that the expected line shape for the Lamb 1057 Mc/s line in hydrogen is Lorentzian within the context of Lamb’s experiments—intensity against Zeeman splitting energy. This is in contradiction to the predictions of Arnous and Heitler. The distribution in k (circular frequency) for a natural line, on the other hand, has a factor k a times the resonant denominator. The definitive position between the conflicting transition rates in the literature is given. The relationship of the calculations to a gauge-independent Hamiltonian, considered in the paper, is investigated in detail. This Hamiltonian, transformed canonically from the conventional one, is shown to be valid for internal, dynamical, quantized electromagnetic fields; its analogue in the case of external applied fields being well known. It is demonstrated that the transformation eliminates many of the difficulties associated with the false near fields and that it also allows certain radiative effects, where the atom acts as a whole, to be computed in a straightforward manner.
The observable line shape of the spontaneous emission depends on the procedure of atom’s excitation. The spectrum of radiation emitted by a two-level atom excited from the ground state by a π pulse of the resonant pump field is calculated for the case when the Rabi frequency is much larger than the relaxation rate. It is shown that the central part of the spectral distribution has a standard Lorentzian form, whereas for detunings from the resonance that are larger than the Rabi frequency the spectral density falls off faster. The shape of the wings of the spectral line is sensitive to the form of the π pulse. The implications for the quantum Zeno effect theory and for the estimates of the duration of quantum jumps are discussed.
The covariant methods of Feynman and Dyson are applied to the problem of line shape. It is shown that the line center depends almost completely on the form of the ${{S}_{\mathrm{Fe}}}^{\ensuremath{'}}$ function, that is the Feynman-type Green's function for the electron interacting with an external field ${{A}_{\ensuremath{\mu}}}^{e}$ and with the quantized radiation field. A finite equation for ${{S}_{\mathrm{Fe}}}^{\ensuremath{'}}$ is derived, from which the line center and shape may be calculated to any desired accuracy. The method is illustrated by two examples: the elastic scattering of photons by a one electron atom in its ground state and the emission of photons by the same type of atom following thermal excitation.
Coherent four-photon scattering of light due to excited and autoionizing states of atomic systems is considered. We analyze the influence of one-photon resonances at the frequency of the anti-Stokes signal and at the frequency of the pump wave on the shape of the spectrum of four-photon scattering. We derive analytical expressions for the line shapes of four-photon scattering involving one-photon resonances with allowance for phase-matching conditions and variation in the populations of excited states for limiting regimes of tight focusing and strong absorption in a medium with a cubic nonlinearity. The calculated spectra of four-photon scattering include the interference of resonant and nonresonant components of the third-order nonlinear susceptibility with allowance for one-photon saturation. We develop a method for normalizing experimental spectra of four-photon scattering with one-photon resonances that takes into account distortion of spectral lines due to the influence of phase mismatch and one- photon absorption.
It is shown that the on-energy-shell single-particle scattering amplitudes obtained from the minimal-coupling and the multipolar forms of the Hamiltonian in non-relativistic quantum electrodynamics are equal to order e 2 ; both forms of the Hamiltonian thus lead to the same differential cross section for Kramers-Heisenberg scattering. The proof given is based directly on the relations which hold between the two transition matrices and is simpler than a previous proof which relied on sum rules derived from the canonical commutation relations. The proof is also more general in that the multipolar Hamiltonian is defined in terms of line integrals along paths which are not assumed to be straight lines but which can be chosen from a large class of possible curves.