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Physical Review A

Elliptical billiard-ball echo model

Raymond John Beach, Burton Brody, Sven R. Hartmann

Published May 1, 1983• 12 citations
Publisher

Abstract

The billiard-ball echo model is expanded to include the effects of using sub-Doppler-linewidth lasers in photon-echo experiments. The simple heuristic model which emerges explains such echo phenomena in gases as the echo time width and the degradation of echo intensity with angling of the excitation pulses. We derive simple results in the small-pulsearea limit and give formal expressions for excitation pulses of arbitrary area.

DOI
10.1103/physreva.27.2537

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Photon echoes, observed recently in ruby and in ${\mathrm{NH}}_{2}$D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.

Physical Review A•May 1, 1976•23 citations
The billiard-ball echo model is expanded to include the effects of using sub-Doppler-linewidth lasers in photon-echo experiments. The simple heuristic model which emerges explains such echo phenomena in gases as the echo time width and the degradation of echo intensity with angling of the excitation pulses. We derive simple results in the small-pulsearea limit and give formal expressions for excitation pulses of arbitrary area.
Photon echoes in gaseous media are explained with the use of a simple heuristic model in which the atoms behave like composite billiard balls. The laser providing the excitation pulses becomes an atom smasher which "splits" the atoms and then by judicious programming puts them back together again. The most general photon-echo reordering process is explained without recourse to formal analysis so that an intuitive feeling is obtained which works equally well for the ordinary two-pulse photon echo, the Raman echo, the trilevel echo, the grating echo, etc. A formal analysis of the billiard-ball echo model is presented in its support.
We have observed the effect of angled excitation beams on the photon echoes from the 22S1/2–22P1/2 transition of atomic lithium. The degradation of the echo from noncollinear excitations is not so great as had been feared. A calculation with the billiard-ball echo model is in good agreement with our experimental results. In a similar experiment in atomic-sodium vapor we have recorded a decay of echo intensity with increasing pulse separation over nearly 12 orders of magnitude, with the weakest signals coming from atoms that had been more than 23 lifetimes in their excited state.
Photon echoes, observed recently in ruby and in NH2{\mathrm{NH}}_{2}NH2​D gas by two-pulse laser excitation, exhibit a modulation behavior as the pulse delay time is varied. The effect, which arises in a three-level quantum system, is due to the interference of two optically coupled transitions that share a common level. In this article, the three-level echo problem is treated exactly. Wave-function solutions are incorporated that are handled conveniently by a Laplace-transform technique. We, therefore, extend the earlier theory of echo modulation by Lambert et al., by removing their restriction of on-resonance pulse excitation. This procedure allows us to examine the effect of inhomogeneous broadening on echo amplitude explicitly, and we treat the problem by numerical and also by analytical means for the case of a Doppler-broadened gas. Analytic expressions are derived either for a high or a low laser power regime corresponding to excitation of either a large or a small fraction of the Doppler linewidth. The former case is analogous to pulsed nuclear magnetic resonance and is well known, whereas the latter result is new and applies to current photon echo measurements.