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Journal of the Optical Society of America B

Lossless propagation of optical pulses through N-level systems with SU(2) symmetry

F. T. Hioe

Published Mar 1, 1989• 13 citations
Publisher

Abstract

Propagation of optical pulses through atomic media consisting of atoms with N transition levels and possessing the so-called SU(2) symmetry is studied. It is shown that there are generally N − 1 sets of conditions, each of which, when satisfied, would permit the appropriate Maxwell–Bloch equations to have a solution having the form of simultaneous different-wavelength optical solitons, so-called simultons. The first two sets of solutions were known previously, but the remaining sets are new.

DOI
10.1364/josab.6.000335

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•Jan 1, 2004•1 citation
Propagation of optical pulses through atomic media consisting of atoms with N transition levels and possessing the so-called SU(2) symmetry is studied. It is shown that there are generally N − 1 sets of conditions, each of which, when satisfied, would permit the appropriate Maxwell–Bloch equations to have a solution having the form of simultaneous different-wavelength optical solitons, so-called simultons. The first two sets of solutions were known previously, but the remaining sets are new.
The propagation of two short different-wavelength optical pulses in three-level absorbers is studied. Combined numerical and analytic techniques are used to solve the three-level Maxwell-Bloch equations that provide the semiclassical description of the problem. The electric field in the model studied consists of two copropagating plane waves, each of which is in near resonance with a transition in the absorber. A new conservation law, holding in the absence of relaxation mechanisms, and independent of particular values of fields and detunings, is given. It is an analog of the law expressing conservation of the Bloch vector length known for two-level atoms. The possibility of simultaneous lossless propagation of the two optical pulses is established. New analytic solutions having the form of simultaneous different-wavelength optical solitons have been found. These pairs of solitons are called simultons. In order for simulton propagation to occur, both the pulses and the medium have to be prepared in a manner determined by the medium's physical parameters. The conditions for such preparation are given. Simultons are predicted to be distinct from the single-wavelength multiple-pulse solutions resulting from large-area-pulse breakup known in two-level absorbers, as well as from two-photon self-induced transparency. Results of numerical experiments on different-wavelength simultaneous propagation are also presented and indicate that simultaneous propagation may also be obtained under less stringent conditions than those predicted by the analytic solutions. Evidence of pulse evolution and breakup is seen.
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.
We introduce a notion of ``matrix potential'' to nonlinear optical systems. In terms of a matrix potential ggg, we present a gauge-field-theoretic formulation of the Maxwell-Bloch equation that provides a semiclassical description of the propagation of optical pulses through resonant multilevel media. We show that the Bloch part of the equation can be solved identically through ggg and the remaining Maxwell equation becomes a second-order differential equation with a reduced set of variables due to the gauge invariance of the system. Our formulation clarifies the (non-Abelian) symmetry structure of the Maxwell-Bloch equations for various multilevel media in association with symmetric spaces G/HG/HG/H. In particular, we associate the nondegenerate two-level system for self-induced transparency with G/H=SU(2)/U(1)G/H=\mathrm{SU}(2)/\mathrm{U}(1)G/H=SU(2)/U(1) and three-level $\ensuremath{\Lambda}$ or VVV systems with G/H=SU(3)/U(2)G/H=\mathrm{SU}(3)/\mathrm{U}(2)G/H=SU(3)/U(2). We give a detailed analysis for the two-level case in the matrix potential formalism, and address various properties of the system including soliton numbers, effective potential energy, gauge and discrete symmetries, modified pulse area, conserved topological, and nontopological charges. The nontopological charge measures the amount of self-detuning of each pulse. Its conservation law leads to a different type of pulse stability analysis that explains earlier numerical results.
A new exact result is presented on the lossless propagation of simultaneous different-wavelength optical pulses through an atomic medium with general NNN energy levels in an extended $\ensuremath{\Lambda}$ or VVV configuration.
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.