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Physical Review A

Coherent propagation and optical pumping in three-level systems

Bruce J. Herman, P. D. Drummond, Joseph H. Eberly, Božena Sobolewska

Published Oct 1, 1984• 10 citations
Publisher

Abstract

We develop a theory to take into account the effects in coherent propagation of a small overlapping of the pump and probe pulses in a three-level system. The Maxwell equations reduce to coupled equations for variables which are related to the time-integrated field envelopes. We make comparisons of the propagation results to the Burnham-Chiao theory for a two-level amplifier and can predict when this theory breaks down. An application to superfluorescence is presented where a lengthening of the delay time is found. We treat Doppler broadening and find simplified equations in two cases: (a) a very broad line and (b) a very narrow line where the initial and final states are degenerate. For the broad-line case, with an appropriate choice of initial parameters, we can make the probe pulse approach any desired area for certain configurations of the levels.

DOI
10.1103/physreva.30.1910

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We develop a theory to take into account the effects in coherent propagation of a small overlapping of the pump and probe pulses in a three-level system. The Maxwell equations reduce to coupled equations for variables which are related to the time-integrated field envelopes. We make comparisons of the propagation results to the Burnham-Chiao theory for a two-level amplifier and can predict when this theory breaks down. An application to superfluorescence is presented where a lengthening of the delay time is found. We treat Doppler broadening and find simplified equations in two cases: (a) a very broad line and (b) a very narrow line where the initial and final states are degenerate. For the broad-line case, with an appropriate choice of initial parameters, we can make the probe pulse approach any desired area for certain configurations of the levels.
A model is presented for the dynamical evolution of superfluorescence from an optically pumped three-level system. The full propagation, transverse, and diffraction effects are taken into account. With the use of a previously developed algorithm, a computational simulation was constructed from this model and results are presented and discussed. In particular, it is shown that the injected coherent pump-pulse initial characteristics, such as on-axis area, temporal and radial width and shape, can have significant deterministic effects on the superfluorescent pulse delay time, peak intensity, temporal width, and shape. Thus, by specifying certain initial properties of the injected pump pulse, the superfluorescent pulse can be shaped and altered. The results predict the conditions under which an injected light pulse of a given frequency can be used to generate, shape, and control a second light pulse of a different frequency via a nonlinear medium, thus demonstrating a new aspect of the phenomenon of light control by light.
An effective two-level Maxwell-Bloch formalism is developed to describe various time-dependent phenomena in a three-level V-type system. The Maxwell-Bloch equations govern the temporal evolution of a probe field while a strong driving field is applied on the adjacent transition. The main assumption adopted throughout the paper is that the spontaneous decay rate on the coherently driven transition exceeds the decay rate on the probe transition as well as inverse time variations of both pulses. It is shown that the probe pulse interacts with such a medium in a manner similar to that in a self-induced transparency. An amplification without inversion of a relatively short probe pulse while propagating through an optically thick atomic medium is analyzed.
We have formulated the quantum theory of formation of the superfluorescence pulse from a pencil-like amplifier. The coupled system of the linearized Bloch equations for two-level atoms and the three-dimensional wave equation for the electric field has been solved in the geometrical-optics approximation for Fresnel numbers $F \ensuremath{\ge} 1$. Several properties of the superfluorescence pulse are discussed. The spatial coherence function of the pulse has been found as a function of the Fresnel number. This function provides a precise formulation of the concept of "diffraction modes." The delay-time distribution function for various Fresnel numbers has been found. The results are consistent with experimental data and numerical simulations.
The coupling of forward and backward propagating waves is studied theoretically in the transition region between superfluorescence (SF) and amplified spontaneous emission (ASE). To accomplish this a computer code has been written that solves Maxwell's equations in time and two spatial dimensions and accounts for counter-propagation, diffraction, and temporal coherence in the Bloch equations. Diffraction is ignored in the present calculations. In the SF regime the coupling is generally weak, as found by others; however, I have found a randomly occurring shot-to-shot coupling that is quite significant in cases where one wave, forward or backward, becomes established from quantum noise first in the amplified rod. In this case the wave that arises second is strongly coupled to the first. In the ASE regime the propagation shows strong forward-backward coupling and the characteristic spatial profile of the inversion that tends to zero at either end and to the small signal value of one in the middle of the amplifier. In the transion region the propagation shows mixed SF and ASE behavior.