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Journal of Applied Physics

Numerical Solutions to a One-Dimensional Analogue of the Diffusion of Resonance Excitation through a Gas

Robert G. McIntyre, Edward M. D'Arcy

Published May 1, 1972• 2 citations
Publisher

Abstract

McIntyre and Fowler reported on an equation for the diffusion of resonance excitation through a gas. A one-dimensional analogue of their equation is presented together with illustrations of numerical solutions. In the case where the mean translational distance moved by an atom between absorption and reemission and the expected skip distance of a photon during an emission-reabsorption process are roughly equal, solutions are obtained considering both processes and then suppressing the former. Illustrations are presented to estimate the relative importance of the contribution of the translation of excited states to the total diffusion process.

DOI
10.1063/1.1661485

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McIntyre and Fowler reported on an equation for the diffusion of resonance excitation through a gas. A one-dimensional analogue of their equation is presented together with illustrations of numerical solutions. In the case where the mean translational distance moved by an atom between absorption and reemission and the expected skip distance of a photon during an emission-reabsorption process are roughly equal, solutions are obtained considering both processes and then suppressing the former. Illustrations are presented to estimate the relative importance of the contribution of the translation of excited states to the total diffusion process.
For a transition in a monatomic gas, it is shown that certain processes in which excitation is transferred from one atom to another lead to different resonance-broadened line shapes in emission and absorption. The absorption line shape is determined by an averaged diagonal matrix element of the resolvent operator (in the representation of the unperturbed states), while in emission some off-diagonal elements contribute. The differences are exhibited formally in terms of resolvent-operator theory, and also worked out explicitly for the case of pure impact broadening with the classical path approximation. In this case, the main effect is a shift of the emission line toward the red, relative to the absorption, by an amount of the same order of magnitude as the width. In the case where the lower level in emission is the one undergoing resonance broadening, the line shapes for emission and absorption (with the same final states) are the same.
The feasibility of resonance transfer of quantum information from one double-level atom to another that is at an arbitrary distance from the former one has been proved. Symmetric and antisymmetric combinations of the wave functions of individual atoms are considered. When taking into account the interatomic dipole-dipole interaction, a certain energy corresponds to each wave function. A solution has been found to a system of equations for the amplitudes of the probability that a resonance photon will be absorbed by one of the system atoms. Three types of solution to the system of equations have been investigated for the resonance and nonresonance absorption of photons and different interatomic distances. It has been shown that when atoms are at an infinite distance from one another, so that there is no dipole-dipole interaction of atoms, quantum information can be transferred from one atom to another via the quantum teleportation channel.
A critical examination of the theory of resonance radiation has been carried through, employing the mathematical technique of Laplace and Stieltjes transforms. Particular attention has been devoted to the question of the temporal behavior of the excited state. With no restrictions on the interaction between atom and radiation field other than that it be real, and that processes of sufficiently high frequency contribute only negligible effects, one can prove that the probability amplitude of the excited state cannot decay according to the general law ${\ensuremath{\Sigma}}_{\ensuremath{\mu},\ensuremath{\nu}}{C}_{\ensuremath{\mu}\ensuremath{\nu}}{t}^{{\ensuremath{\lambda}}_{\ensuremath{\mu}}}{e}^{\ensuremath{-}{\ensuremath{\beta}}_{\ensuremath{\nu}}t},$ where the ${\ensuremath{\lambda}}_{\ensuremath{\mu}}$ and ${\ensuremath{\beta}}_{\ensuremath{\nu}}$ are complex constants lying in the right half-plane and the ${C}_{\ensuremath{\mu}\ensuremath{\nu}}$ are arbitrary complex coefficients. The deviation from the law just cited can be termed a straggling phenomenon since exact analysis shows that the probability amplitude, for sufficiently long times, is greater than that defined according to the radioactive decay law.Rigorous analysis of the source of this apparently anomalous behavior reveals the following explanation. Although the probability amplitude of the excited state is, strictly speaking, a functional of the interaction between the transition current in the atom and the photon states of the electromagnetic field, the essential features are determined principally by those states lying near the resonance frequency. Three elementary observations are immediately apparent. First, a particularly tractable analytical approximation for the interaction can be made such that the value of the actual interaction is reproduced at the resonance frequency and such that the behavior for high and low frequency photon states is at least qualitatively correct. Exact solutions in closed form can be obtained for this interaction. Second, in terms of the preceding representation it is possible to show that, associated with the transition between two atomic states, one is presented with a concomitant picture of a damped oscillation of charge describable in purely classical terms. Third, this classical motion can be interpreted as forming a source function for a continuous stochastic process in terms of which one finally derives the representation of the probability amplitude of the excited state. The straggling phenomenon previously cited is therefore that associated with all diffusion processes. The consideration of the exact interaction leads to more involved diffusion phenomena but does not in any case permit a radioactive decay law.
We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 \AA{}, respectively.
A formalism introduced previously by the author is used to calculate the shape of an absorption line in a gas, taking account of the "resonance" interaction between like atoms. The most serious approximation is the neglect of the effect of translational motion. An asymmetric line is obtained, with the maximum shifted slightly toward the red. Some simple classical considerations also agree qualitatively with the results. Various properties of the line shape obtained are briefly discussed. The neglect of translational motion makes it impossible to compare the results with experiments performed up to the present time.