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Physical Review A

Natural line shape

Peter W. Milonni, R. J. Cook, Jay R. Ackerhalt

Published Oct 1, 1989• 23 citations
Publisher

Abstract

For many years there have been two expressions for the natural radiative line shape, depending on the form of the Hamiltonian assumed for the interaction of an atom with the radiation field. It is shown that this difference arises from artificial assumptions about the atom-field interaction, and that the same line shape is obtained from either form under the appropriate physical assumptions.

DOI
10.1103/physreva.40.3764

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For many years there have been two expressions for the natural radiative line shape, depending on the form of the Hamiltonian assumed for the interaction of an atom with the radiation field. It is shown that this difference arises from artificial assumptions about the atom-field interaction, and that the same line shape is obtained from either form under the appropriate physical assumptions.
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.
We use a general formulation of nonrelativistic quantum electrodynamics in which the gauge freedom is carried by the arbitrary transverse component of the the Green’s function for the divergence operator to calculate the natural lineshape of spontaneous emission, thus discerning the full dependence of the result on the choice of gauge. We also use a representation of the Hamiltonian in which the virtual field associated with the atomic ground state is explicitly absent. We con-sider two processes by which the atom is excited; the first is resonant absorption of incident radiation with a sharp line. This treatment is then adapted to derive a resonance fluorescence rate associated with the Lamb line in atomic hydrogen. Sec-ond we consider the atom’s excitation due to irradiation with a laser pulse treated semi-classically. An experiment could be used to reveal which of the calculated line-shape distributions is closest to the measured one. This would provide an answer to a question of fundamental importance; how does one best describe atom-radiation interactions with the canonical formalism? 1
The resonance fluorescence of a system of two identical, interacting atoms is investigated when the atoms are separated by distances that are large compared with atomic size. We consider the interactions between the atoms and the radiation field and the electrostatic interaction between the atoms, and find that the line shape of the emitted radiation consists of two peaks whose amplitude, separation, and width are functions of the internuclear distance RRR of the atoms. The peaks are distinct at smaller separations, and, as RRR increases, rapidly merge to the characteristic line shape of a single, isolated atom.
The processes of absorption and emission of radiation by a system of two interacting atoms is investigated. Separation between the atoms is assumed to be much larger than the atomic diameter, and interactions considered are those between the individual atoms and the radiation field and the electrostatic interaction between the atoms. Calculations include those for the time-dependent probability amplitude of the intermediate and final states as well as that of the emission line shape. The line shape of the emitted radiation consists of two peaks whose amplitude, separation, and half-width are functions of both the internuclear separation R of the atoms and the energy difference of the excited states of the two atoms. For a large energy difference, the peaks are distinct; for small energy differences, the peaks are distinct at smaller separations, but merge toward the line shape characteristic of a single, isolated atom as R increases.
We use a general formulation of nonrelativistic quantum electrodynamics in which the gauge freedom is carried by the arbitrary transverse component of the the Green's function for the divergence operator to calculate the natural lineshape of spontaneous emission, thus discerning the full dependence of the result on the choice of gauge. We also use a representation of the Hamiltonian in which the virtual field associated with the atomic ground state is explicitly absent. We consider two processes by which the atom is excited; the first is resonant absorption of incident radiation with a sharp line. This treatment is then adapted to derive a resonance fluorescence rate associated with the Lamb line in atomic hydrogen. Second we consider the atom's excitation due to irradiation with a laser pulse treated semi-classically. An experiment could be used to reveal which of the calculated lineshape distributions is closest to the measured one. This would provide an answer to a question of fundamental importance; how does one best describe atom-radiation interactions with the canonical formalism?
The relationship between the multipolar and minimal Hamiltonians through a canonical transformation is used to analyze the time dependence of quantum operators in the two formalisms. It is shown that operators dependent on particle position, the vector potential, and their time derivatives have the same time dependence. However, operators such as the photon number and the atomic population number evolve differently in the two cases. Expressions correct to second order in the electric-dipole moments for these are given. The expectation values of the photon number operator are calculated in the two formalisms and are used to predict natural line shapes. These theoretical shapes differ. The observed shape will depend on the particular setup of the experiment involving the radiative decay of an excited atom. A comparison with the theoretical predictions will determine which of the two frameworks is most appropriate to describe the decay. Finally, the energy density of the electromagnetic field in the neighborhood of an atom is calculated within the two formalisms.