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arXiv:solv-int/9904007

Complex sine-Gordon Equation in Coherent Optical Pulse Propagation

Q‐Han Park, Hyung‐Joon Shin

Published Apr 6, 1999• 3 citations
arXivHTMLPDF

Abstract

It is shown that the McCall-Hahn theory of self-induced transparency in coherent optical pulse propagation can be identified with the complex sine-Gordon theory in the sharp line limit. We reformulate the theory in terms of the deformed gauged Wess-Zumino-Witten sigma model and address various new aspects of self-induced transparency.
DOI
10.48550/arxiv.solv-int/9904007

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A systematic perturbation expansion with respect to the small parameter ${(\ensuremath{\Delta}\ensuremath{\tau}\ensuremath{\omega})}^{\ensuremath{-}1}$, $\ensuremath{\Delta}\ensuremath{\tau}$ being the duration of the pulse and $\ensuremath{\omega}$ the optical frequency, is introduced. This method is applied to the problem of self-induced transparency for sharp-line absorbers at exact resonance. Corrections to the McCall and Hahn solution are obtained which describe the propagation of the form-preserving pulse with modulated frequency and slightly changed shape, group velocity, and wave vector. Earlier results by Marth, Holmes, and Eberly on phase modulation and wave-vector shift are confirmed.
Hidden nonabelian symmetries in nonlinear interactions of radiation with matter are clarified. In terms of a nonabelian potential variable, we construct an effective field theory of self-induced transparency, a phenomenon of lossless coherent pulse propagation, in association with Hermitian symmetric spaces G/HG/HG/H. Various new properties of self-induced transparency, e.g. soliton numbers, effective potential energy, gauge symmetry and discrete symmetries, modified pulse area, conserved U(1)U(1)U(1)-charge etc. are addressed and elaborated in the nondegenerate two-level case where G/H=SU(2)/U(1)G/H = SU(2)/U(1)G/H=SU(2)/U(1). Using the U(1)U(1)U(1)-charge conservation, a new type of analysis on pulse stability is given which agrees with earlier numerical results.
A review is given of the current status of the self-induced transparency effect. The McCall–Hahn theory (2π pulses) is considered and its generalization to the case of phase modulation of radiation pulses, degeneracy of transitions, and two-photon resonance is discussed. The relationship with the theory of nonlinear waves, obeying the Korteweg–de Vries equation, is pointed out. The possibility of similar coherent effects in direct interband transitions in semiconductors is discussed. A summary is given of the experimental results demonstrating a strong fall in the absorption and a considerable slowing down of light pulses, typical of the self-induced transparency. Possible physical applications of the effect, based on these properties, are discussed.
A review is given of the investigations of the coherent interaction of light pulses with resonantly absorbing two-level media. An analysis is made of the self-induced transparency discovered by McCall and Hahn and of its importance in quantum electronics. Bleached states associated with the phase modulation are considered in detail. A comparison is made with the theory of nonlinear waves of different physical origin. A review is given of some possible physical applications associated with the discriminatory nature of the transmission and delay of light in the self-induced transparency effect.
The propagation of ultrashort coherent light pulses through a resonant atomic medium, modelled by a two-quantum level of absorbers, is studied including the effects of level degeneracy. The governing equations, a sequence of n-tuple sine-Gordon (SG) equations, are solved by quadrature for the Q-transition, with an approximate method indicated for the (mathematically) more difficult P- and R- transitions. It is shown, in particular, that there is a surprisingly close relation between the usual SG equation and the double SG equation. A simple theorem is proved which shows that each travelling wave solution of the former immediately leads to a solution of the latter, which thus admits a restricted Backlund transformation. This direct solution of nonlinear dispersive equations, which apparently cannot be solved by the inverse method, has an interesting by-product: an approximate method for finding travelling wave solutions of relatively arbitrary Klein-Gordon equations.