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Bulletin of the Russian Academy of Sciences Physics

Nonlinear Dynamics of Extremely Short Solitons of the Generalized Reduced Maxwell–Bloch System

Sergey Vladimirovich Sazonov, N. V. Ustinov

Published Jan 1, 2020• 4 citations
Publisher

Abstract

Abstract The effect of remote quantum states on two allocated atomic levels during the propagation of extremely short electromagnetic pulses in a medium is considered without using the slowly varying envelope approximation. A generalization of the reduced Maxwell–Bloch system is obtained that is integrable via the inverse scattering transformation method. Its soliton and breather solutions are found and studied.

DOI
10.3103/s1062873820010244

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Abstract The effect of remote quantum states on two allocated atomic levels during the propagation of extremely short electromagnetic pulses in a medium is considered without using the slowly varying envelope approximation. A generalization of the reduced Maxwell–Bloch system is obtained that is integrable via the inverse scattering transformation method. Its soliton and breather solutions are found and studied.
The physical generalization of the system of the reduced Maxwell-Bloch equations describing the nonlinear interaction of the laser pulses with the multilevel quantum medium is obtained without using the slowly varying envelope approximation. It is shown that this system is integrable in the frameworks of the inverse scattering transformation method. The soliton and breather solutions in the form of the unipolar and few-cycle pulses are constructed. Their distinctive features caused solely by the multilevel structure of the medium are discussed. It is revealed that the collision of the solitons can lead to an appearance of the large-amplitude short-living pulse, whose dynamics resembles that of rogue waves.
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.
Effects of propagation of an extremely short (of one or several oscillation periods) electromagnetic pulse in a medium whose resonance transition is characterized by diagonal as well as nondiagonal matrix elements of the dipole moment operator have been studied numerically. The Maxwell-Bloch system of equations is employed without using the approximation of slowly varying envelopes. An analog of the McCall and Hahn area theorem is discussed as applied to the division of the initial extremely short pulse into subpulses. The solution is obtained in the form of a solitary stable bipolar signal with a nonzero pulse area (nonzero breather).
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.