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Inverse Problems

Darboux transformations and coherent interaction of the light pulse with two-level media

Vladimir Borisovich Matveev, Mikhail A. Salle, Andrei V. Rybin

Published Feb 1, 1988• 31 citations
Publisher

Abstract

Applications of the Darboux transformation method to study the reduced Maxwell-Bloch (RMB) system and of the self-induced transparency (SIT) equations are considered. Both systems describe the propagation of ultrashort optical pulses in a two-level medium to a reasonable approximation. The main result of the present work is the construction of multisoliton solutions on an arbitrary background for RMB and SIT equations. Particular cases of these solutions are discussed in some detail.

DOI
10.1088/0266-5611/4/1/015

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(Submitted 14 February 1986) Zh. Eksp. Teor. Fiz. 92,46-55 (January 1987) The coherent resonant interaction between multifrequency light and a multilevel system is considered together with the N-wave interaction of wave packets in a medium with a quadratic nonlinearity. The Lax representation of the self-consistent set of Maxwell-Bloch equations for this interaction is found. Soliton solutions resulting from the parametric transformation of ultrashort pulses in three- and four-level system (N = 3, and N

Zhurnal Eksperimentalnoi i Teoreticheskoi Fiziki•Jan 1, 1987•0 citations
Applications of the Darboux transformation method to study the reduced Maxwell-Bloch (RMB) system and of the self-induced transparency (SIT) equations are considered. Both systems describe the propagation of ultrashort optical pulses in a two-level medium to a reasonable approximation. The main result of the present work is the construction of multisoliton solutions on an arbitrary background for RMB and SIT equations. Particular cases of these solutions are discussed in some detail.
The phenomenon that describes the effect of a coherent medium response to an incident electric field, to which the medium is totally transparent and which undergoes lossless propagation, is known as self-induced transparency (SIT). SIT was first discovered by McCall and Hahn (1969) in the case of nondegenerate two-level atoms. Special solutions for the two-level system were found by Lamb (1971), while the initial value problem for the propagation of a pulse through a resonant two-level optical medium was solved by Inverse Scattering Transform (IST) in [1, 2]. It is possible to formulate the SIT equations in the framework of the IST also in the case of a three-level system, as in [3]. While the associated scattering problem is the same as for the coupled nonlinear Schrödinger equation, the time evolution depends on asymptotic values of the material polarizability envelopes and is highly non-trivial. This talk will address the solution of the initial value problem for the SIT equations for three level systems, for generic preparation of the medium, and
The coherent propagation of two optical short pulses through a resonant medium consisting of two three-level atoms in the configuration is investigated. A self-consisting analytical solution without steady state or adiabatic approximations is presented. The electric field in the model studied consists of two co-propagating plane waves, each of which is in near resonance with a transition in the absorber. The density matrix of two three level atoms is studied. Also, the reduced density matrices are stated. The present approach for the semiclassical treatment of resonant coherent interactions in three-level atoms represents a generalization of the Maxwell- Bloch equations for a two-level system. The Maxwell-Bloch equations reduced to the non-linear pendulum equation. The solution shows that the two pulses can propagate simultaneously without loss under some conditions about the pulses and the medium. The propagation of the two pulses through the medium without any constrains approves the self -induced transparency (SIT) phenomenon.
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.
The form invariant coherent pulse propagation in semiconductors excited at 1s-exciton resonance is studied analytically using the reduced semiconductor Maxwell-Bloch equations. The sech-shaped pulse solution for excitonic self-induced transparency (SIT) is presented, showing significant difference in comparison with the well known SIT solution for non-interacting two-level systems. In contrast to 2π pulses in atomic systems, the phenomenon of SIT of interacting excitons in semiconductors occurs for the pulses of area 1.07π. Possible applications of the SIT solitons in semiconductor all-optical switching devices are discussed.
We reveal the existence of an optical self-induced-transparency soliton family in a two-level absorbing system down to the few-cycle limit. Based on the few-cycle envelope approximation, we introduce a systematical method of reducing the Maxwell-Bloch equations beyond the slowly varying envelope approximation and characterize the parameter space to achieve slow and fast lights. Verified by direct numerical simulations with full vectorial Maxwell-Bloch equations, we also give the relationships for a number of optical cycles on area theory and pulse group velocity, which demonstrate possible coherent photon-matter interactions.
(Submitted 14 February 1986) Zh. Eksp. Teor. Fiz. 92,46-55 (January 1987) The coherent resonant interaction between multifrequency light and a multilevel system is considered together with the N-wave interaction of wave packets in a medium with a quadratic nonlinearity. The Lax representation of the self-consistent set of Maxwell-Bloch equations for this interaction is found. Soliton solutions resulting from the parametric transformation of ultrashort pulses in three- and four-level system (N = 3, and N