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Physical Review A

Matter-field interaction in atomic physics and quantum optics

Willis E. Lamb, Rainer R. Schlicher, Marlan O. Scully

Published Sep 1, 1987• 146 citations
Publisher

Abstract

Almost forty years ago one of us (W.E.L.) noticed that within the analysis of experiments on the hydrogen fine structure the matter-field interaction was sensitive to the choice of gauge when decaying states were used. In the present paper this problem is resolved for the Bethe-Lamb equations in the context of the two-level model with Weisskopf-Wigner decay included.

DOI
10.1103/physreva.36.2763

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Almost forty years ago one of us (W.E.L.) noticed that within the analysis of experiments on the hydrogen fine structure the matter-field interaction was sensitive to the choice of gauge when decaying states were used. In the present paper this problem is resolved for the Bethe-Lamb equations in the context of the two-level model with Weisskopf-Wigner decay included.
It is shown that the application of a self-consistent projection-operator method, developed recently by the author, makes possible a complete and gauge-invariant treatment of spontaneously decaying atomic state. All the shortcomings of the Weisskopf-Wigner gauge-dependent method (Markov approximation and the neglect of virtual transition, retardation and electron-spin effects) are removed. After deriving a gauge-invariant equation of motion for the decaying state, a new physical interaction picture (in which the unobservable interaction of the free electron with the vacuum radiation field is eliminated) is introduced. In this new interaction picture, apart from a natural solution of the renormalization problem, explicit finite analytic results for the whole non-Markovian Lyman- alpha spontaneous decay are derived. Furthermore, a quasi-Lorentzian expression for the radiative line shape containing the correct (complete) finite non-relativistic frequency shift, stemming from the Lamb shifts of both levels 1S and 2P, is calculated.
For the first time to the authors' knowledge a mathematically rigorous method is used for treating finite-time deviations from the exponential decay in the case of spontaneous Lyman-\ensuremath{\alpha} transitions in a two-level hydrogenic atom. First, in the so-called Weisskopf-Wigner model (where the rotating-wave approximation is implied) finite-time deviations with a rigorous validity range, based on accurate error estimations, are derived. Second, in order to obtain the frequency shift (Lamb shift), counter-rotating terms are taken into account by using a projection-operator method developed a decade ago by one of the present authors [J. Seke, Phys. Rev. A 21, 2156 (1980)]. By means of this method, equations of motion for dipole-moment expectation values are derived without employing the usual Born approximation. These equations are solved by using the method of the Laplace transform and its inverse. Again finite-time deviations from the exponential decay of the dipole-moment expectation values with a definite validity range are obtained. Finally, it should be emphasized that all the results presented in this paper contain accurate error estimates which are absolutely necessary in a rigorous mathematical method.
Protons with velocities ranging from 3.2 \ifmmode\times\else\texttimes\fi{} 108{10}^{8}108 to 6.3 \ifmmode\times\else\texttimes\fi{} 108{10}^{8}108 cm/sec were passed through a carbon foil producing excited hydrogen atoms which decayed in flight after emergence. With the foil in magnetic fields from 2 to 25 G, the lines ${\mathrm{H}}_{\ensuremath{\alpha}}$ through ${\mathrm{H}}_{\ensuremath{\epsilon}}$ showed periodic intensity fluctuations due to Stark mixings of fine-structure levels caused by motional electric fields (6-158 V/cm). An extension of the Bethe-Lamb theory of Stark quenching provides good agreement between a small number of theoretically dominant frequencies and the measured fluctuation frequencies. Apparent discrepancies between theory and experiment noted by other experimenters are shown to be removed by our calculations except in one measurement on ${\mathrm{Ly}}_{\ensuremath{\alpha}}$. The necessity of using double foils postulated in the previous work is discussed. Our analysis suggests the preferential alignment of the final-state angular momenta perpendicular to the beam direction. Reasons for this alignment are given.