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arXiv:solv-int/9709003 · Physical Review A

Matched pulse propagation in a three-level system

Q‐Han Park, Hyung‐Joon Shin

Published Jun 1, 1998• 57 citations
arXivHTMLPDF

Abstract

The B\"acklund transformation for the three-level coupled Schr\"odinger-Maxwell equation is presented in the matrix potential formalism. By applying the B\"acklund transformation to a constant-electric-field background, we obtain a general solution for matched pulses (a pair of solitary waves) that can emit or absorb a light velocity solitary pulse but otherwise propagate with their shapes invariant. In the special case, this solution describes a steady-state pulse without emission or absorption, and becomes the matched pulse solution recently obtained by Hioe and Grobe [Phys. Rev. Lett. 73, 2559 (1994)]. A nonlinear superposition rule is derived from the B\"acklund transformation and used for the explicit construction of two solitons as well as non-Abelian breathers. Various features of these solutions are addressed. In particular, we analyze in detail the scattering of ``binary solitons,'' a specific pair of different wavelength solitons, one of which moves with the velocity of light. Unlike the usual case of soliton scattering, the broader soliton changes its sign after the scattering, thus exhibiting a binary behavior. Surprisingly, the light velocity soliton receives a time advance through the scattering, thereby moving faster than light, which, however, does not violate causality.

DOI
10.1103/physreva.57.4643

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The B"acklund transformation for the three-level coupled Schr"odinger-Maxwell equation is presented in the matrix potential formalism. By applying the B"acklund transformation to a constant-electric-field background, we obtain a general solution for matched pulses (a pair of solitary waves) that can emit or absorb a light velocity solitary pulse but otherwise propagate with their shapes invariant. In the special case, this solution describes a steady-state pulse without emission or absorption, and becomes the matched pulse solution recently obtained by Hioe and Grobe [Phys. Rev. Lett. 73, 2559 (1994)]. A nonlinear superposition rule is derived from the B"acklund transformation and used for the explicit construction of two solitons as well as non-Abelian breathers. Various features of these solutions are addressed. In particular, we analyze in detail the scattering of ``binary solitons,'' a specific pair of different wavelength solitons, one of which moves with the velocity of light. Unlike the usual case of soliton scattering, the broader soliton changes its sign after the scattering, thus exhibiting a binary behavior. Surprisingly, the light velocity soliton receives a time advance through the scattering, thereby moving faster than light, which, however, does not violate causality.
Exact soliton solutions of a coupled system of Maxwell-Bloch equations are obtained analytically within the inverse scattering transform scheme. The equations describe a light pulse coherently propagating through a mixed ensemble of two-level atoms characterized by two slightly different detuning parameters. The solution illustrates the information about the detailed atomic properties which are impressed on both phase and velocity of the pulse during propagation, with potential applications in high resolution spectroscopy.
The dynamics of soliton pulses for use in nonlinear optical devices is mathematically modelled by Maxwell-Bloch systems of equations for the interaction of light with a uniform distribution of quantum-mechanical atoms. We study the Reduced Maxwell-Bloch (RMB) equations occurring when an ensemble of rotationally symmetric 3-level atoms is assumed. The model applies for on and off-resonance conditions and is completely integrable using Inverse Scattering theory, since it arises as the compatibility condition of a 3 x 3 AKNS-system. Furthermore this integrability remains valid for all timescales of the optical field because only the “one-way wave approximation” is required during the derivation. Solutions are constructed in two ways: 1. Darboux-Bäcklund transforms are applied, generating single soliton pulses of ultrashort (< 1ps) duration, and families of elliptically polarised 2-solitons not possible in lower dimensional problems. 2. A general Inverse Scattering scheme is developed and tested. The Direct Scattering Problem is dealt with first to obtain a complete set of scattering data. Subsequently the Inverse Problem is solved both formally and then in explicit closed form for the special case that the reflection coefficients vanish for real values of the spectral parameter. In this case the main result is a determined system of n linear algebraic equations which yield the n-soliton of our RMB-system. It is confirmed that the 1-solitons found by means of Darboux transform are precisely the same as those given by the full mechanism of Inverse Scattering. Finally we calculate the invariants of the motion for the RMB-equations, and derive an evolution equation giving the variation with propagation distance of the invariant functionals when the original RMB-system is modified by an arbitrary perturbing term. As an application dissipative effects on 1-solitons are considered.
One-dimensional pulse propagations in the electromagnetically induced media are studied. The Maxwell–Bloch equations for a three-level medium are simplified with the approximations of slowly varying amplitudes, negligible irreversible relaxations and exact resonances. An analysis is made for the case where the integrable condition is satisfied; the oscillation strengths for the two transitions are equall. Through the Bäcklund transformation, one-soliton solutions including the so-called simulton are derived.