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Physical Review A

Some Results on Coherent Radiative Phenomena with 0π Pulses

Frederic A. Hopf, George L. Lamb, C. K. Rhodes, Marlan O. Scully

Published Feb 1, 1971• 46 citations
Publisher

Abstract

The propagation behavior of nontrivial $0\ensuremath{\pi}$ pulses [$\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\mathcal{E}(t,z)dt=0$, $\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}{\mathcal{E}}^{2}(t,z)\mathrm{dt}>0$] in attenuating media is examined. It is found that under appropriate conditions $0\ensuremath{\pi}$ pulses exhibit anomalously low absorption which is a direct consequency of atomic coherence. Specific analytical results valid for an unbroadened medium are presented, while the corresponding solutions for an inhomogeneously broadened resonance are calculated numerically. These two results are consistent in the limit of ${T}_{2}^{*}\ensuremath{\rightarrow}\ensuremath{\infty}$. Furthermore, the influences of ${T}_{2}$, ${T}_{2}^{*}$, and level degeneracy on the characteristics of propagation are determined. Finally, the behavior of more general $0\ensuremath{\pi}$ pulses is studied. The criteria for the production of $0\ensuremath{\pi}$ pulses are qualitatively established by numerical methods, and the influence of the relative phase of pulse pairs on propagation is examined. The latter indicates a method for the measurement of relative pulse phase.

DOI
10.1103/physreva.3.758

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The propagation behavior of nontrivial 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses [\ensuremath∫\ensuremath−\ensuremath∞\ensuremath∞E(t,z)dt=0\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}\mathcal{E}(t,z)dt=0\ensuremath∫\ensuremath−\ensuremath∞\ensuremath∞E(t,z)dt=0, \ensuremath∫\ensuremath−\ensuremath∞\ensuremath∞E2(t,z)dt>0\ensuremath{\int}{\ensuremath{-}\ensuremath{\infty}}^{\ensuremath{\infty}}{\mathcal{E}}^{2}(t,z)\mathrm{dt}>0\ensuremath∫\ensuremath−\ensuremath∞\ensuremath∞E2(t,z)dt>0] in attenuating media is examined. It is found that under appropriate conditions 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses exhibit anomalously low absorption which is a direct consequency of atomic coherence. Specific analytical results valid for an unbroadened medium are presented, while the corresponding solutions for an inhomogeneously broadened resonance are calculated numerically. These two results are consistent in the limit of T2∗\ensuremath→\ensuremath∞{T}_{2}^{*}\ensuremath{\rightarrow}\ensuremath{\infty}T2∗​\ensuremath→\ensuremath∞. Furthermore, the influences of T2{T}_{2}T2​, T2∗{T}_{2}^{*}T2∗​, and level degeneracy on the characteristics of propagation are determined. Finally, the behavior of more general 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses is studied. The criteria for the production of 0\ensuremathπ0\ensuremath{\pi}0\ensuremathπ pulses are qualitatively established by numerical methods, and the influence of the relative phase of pulse pairs on propagation is examined. The latter indicates a method for the measurement of relative pulse phase.
The initial value problem for the propagation of a pulse through a resonant two-level optical medium is solved by the inverse scattering method. In general, an incident pulse decomposes not only into a special class of pulses to which the medium is transparent but also yields radiation which is absorbed by the medium. In this respect ``this problem'' has properties markedly different from other dispersive and reversible wave phenomena some of which are tractable by the inverse scattering method. Indeed, it is remarkable that in the present case the method still applies. In particular, we show that, while there are an infinite number of local conservation laws, the integrated densities, and in particular the energy, are only conserved for a very special class of initial conditions. The theoretical results obtained are in close agreement with all the qualitative features observed in the experiments on coherent pulse propagation. Finally, we also show that causality is preserved. Two new and novel features are introduced and briefly discussed. First, we show that if the homogeneous broadening effect is a function of position in the medium, the pulses may speed up and slow down accordingly, without losing their permanent identities. Second, we have found a new kind of solution mode corresponding to a proper eigenvalue of the scattering problem which is not a bound state.
A theoretical model for the propagation of coherent light pulses in resonant media where a large number of atoms cooperate in spontaneous emission (superradiance) is discussed. The medium is supposedly free of inhomogeneous broadening, and no lossy relaxation mechanisms are taken into account. The conditions for the emission of all spontaneous radiation in the direction of pulse propagation are described. A semi-classical model is used and, at each point in the medium, the propagation field and the spontaneous damping field are considered separately. The equation governing the evolution of a superradiant system in the presence of an applied pulse is derived. A mechanism of energy transfer from the traveling pulse to the medium is described which includes superradiant spontaneous emission effects. As a result of the neglect of all lossy relaxation processes, all energy stored in the medium is eventually returned to the pulse by coherent spontaneous emission. Therefore, in this theory, all pulses travel through the medium without energy loss. Numerical results for square input pulses are discussed which show two possible types of propagation behavior: (a) Low input intensities produce a negative tail to the input (positive) pulse and sharpening of the tail with the advance in the medium; (b) higher input intensities produce, after sufficient penetration, a steady-state pulse envelope somewhat similar to the $2\ensuremath{\pi}$ pulses of self-induced transparency. The steady-state solutions of the equations of motion are found numerically and the pulse shapes are computed for different degrees of superradiance. In the limit of strong superradiance, very asymmetric pulse envelopes of large area are found. For negligible superradiance, approximately $2\ensuremath{\pi}$ hyperbolic-secant pulses are obtained. The effects of inhomogeneous broadening are briefly considered, and the consequences of important superradiant damping in photon-echo experiments are qualitatively discussed.
The influence of degeneracy on coherent pulse propagation in an inhomogeneously broadened attenuator is examined by computer calculations, and the connection with self-induced transparency in a nondegenerate system is described. Specific modifications that arise in a degenerate system are the development of pulse shapes characteristic of the degeneracy, the association of a finite loss with propagation, a reduction of pulse delay times, and the suppression of pulse separation of multi-$2\ensuremath{\pi}$ pulses. Additionally, results of calculations pertaining to experiments on pulse propagation in SF6{\mathrm{F}}_{6}F6​ are presented which indicate two basically different phenomena. One corresponds to delay phenomena qualitatively similar to nondegenerate self-induced transparency, while the other is a coherent saturation effect generally characteristic of degenerate media. The computed results fully substantiate our earlier experimental findings.
This article presents a study of the propagation of small-area pulses of coherent light through matter. It is found that the small-area pulses exactly obey an area theorem which, in the case of an attenuating medium, requires that the pulse area decay to zero exponentially with increasing distance of propagation. This, however, does not necessarily imply that the pulse energy decays exponentially. Instead, it is found that pulses of duration comparable to or shorter than the transverse relaxation time of the medium T2{T}_{2}T2​ (including both homogeneous and inhomogeneous broadening) propagate with low energy loss. The results are explained in terms of simple physical arguments which indicate that the pulse envelope should oscillate between positive and negative values, causing the area to decrease without a comparable decrease in the pulse energy. Analytic solutions are presented for the case of a pulse whose envelope varies as ${t}^{k}{e}^{\ensuremath{-}t}$ for tg0tg0tg0, a rectangular pulse, an ultrashort pulse (i.e., pulse width much less than T2{T}_{2}T2​), and the leading portion of a step-function pulse. The propagation of a small-area Gaussian pulse is studied numerically.
This article presents a study of the propagation of small-area pulses of coherent light through matter. It is found that the small-area pulses exactly obey an area theorem which, in the case of an attenuating medium, requires that the pulse area decay to zero exponentially with increasing distance of propagation. This, however, does not necessarily imply that the pulse energy decays exponentially. Instead, it is found that pulses of duration comparable to or shorter than the transverse relaxation time of the medium T2{T}_{2}T2​ (including both homogeneous and inhomogeneous broadening) propagate with low energy loss. The results are explained in terms of simple physical arguments which indicate that the pulse envelope should oscillate between positive and negative values, causing the area to decrease without a comparable decrease in the pulse energy. Analytic solutions are presented for the case of a pulse whose envelope varies as ${t}^{k}{e}^{\ensuremath{-}t}$ for t>0t>0t>0, a rectangular pulse, an ultrashort pulse (i.e., pulse width much less than T2{T}_{2}T2​), and the leading portion of a step-function pulse. The propagation of a small-area Gaussian pulse is studied numerically.
An analytic solution is obtained for the self-consistent equations describing propagation of an ultrashort pulse in a resonant two-level medium. The first approximation in respect of the dispersion is used to allow for phase modulation and for the dependence of the group velocity on the intensity and on the running value of the frequency. The conditions for the formation of a shock wave are established. It is shown that asymmetric broadening of the spectrum of a pulse results in coherent absorption, as a result of which the propagation of this pulse in such a medium leaves behind a polarization at the atomic transition frequency responsible for the radiation represented by the pulse.