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Physical Review A

Slow-light solitons in two-level media generated by evanescent fields

Jérôme J.-P. Leon

Published Jun 12, 2007• 6 citations
Publisher

Abstract

Coupling a two-level medium, described by the Maxwell-Bloch (MB) equations, to an external cavity mode results in a prescribed boundary value for the electric field that can be tuned to the forbidden band gap of the medium. It is shown that there exists a threshold of energy density flux above which a solitonlike light pulse is generated and propagates in the medium at a fraction of the light velocity. When the driving frequency becomes close to the band gap edge, where the group velocity vanishes, the MB equations tend to the nonlinear Schr\"odinger equation which furnishes the theoretical ground to understand soliton generation by evanescent fields as a manifestation of nonlinear supratransmission, produced by an instability of the evanescent wave. Moreover, the propagation at a fraction of light velocity is shown to result from a continuous periodic exchange between polarization (electromagnetic energy) and population (density of atoms in the excited level). The process is demonstrated to occur at twice the internal frequency of the generated gap soliton. Last, when the medium spatial extension is of the order of the soliton dimension, an optical bistability is shown to occur at the supratransmission threshold allowing for different output intensities corresponding to a given input value.

DOI
10.1103/physreva.75.063811

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Coupling a two-level medium, described by the Maxwell-Bloch (MB) equations, to an external cavity mode results in a prescribed boundary value for the electric field that can be tuned to the forbidden band gap of the medium. It is shown that there exists a threshold of energy density flux above which a solitonlike light pulse is generated and propagates in the medium at a fraction of the light velocity. When the driving frequency becomes close to the band gap edge, where the group velocity vanishes, the MB equations tend to the nonlinear Schr"odinger equation which furnishes the theoretical ground to understand soliton generation by evanescent fields as a manifestation of nonlinear supratransmission, produced by an instability of the evanescent wave. Moreover, the propagation at a fraction of light velocity is shown to result from a continuous periodic exchange between polarization (electromagnetic energy) and population (density of atoms in the excited level). The process is demonstrated to occur at twice the internal frequency of the generated gap soliton. Last, when the medium spatial extension is of the order of the soliton dimension, an optical bistability is shown to occur at the supratransmission threshold allowing for different output intensities corresponding to a given input value.
The Maxwell–Bloch system describes a quantum two-level medium interacting with a classical electromagnetic field by mediation of the population density. This population density variation is a purely quantum effect which is actually at the very origin of nonlinearity. The resulting nonlinear coupling possesses particularly interesting consequences at the resonance (when the frequency of the excitation is close to the transition frequency of the two-level medium) as e.g. slow-light gap solitons that result from the nonlinear instability of the evanescent wave at the boundary. As nonlinearity couples the different polarizations of the electromagnetic field, the slow-light gap soliton is shown to experience effective scattering with charges in the medium, allowing it for instance to be trapped or reflected. This scattering process is understood qualitatively as being governed by a nonlinear Schrödinger model in an external potential related to the charges (the electrostatic permanent background component of the field).
We consider the continuous-wave (cw) propagation through a fiber Bragg grating that is uniformly doped with two-level resonant atoms. Wave propagation is governed by a system of nonlinear coupled-mode Maxwell-Bloch (NLCM-MB) equations. We identify modulational instability (MI) conditions required for the generation of ultrashort pulses in both anomalous and normal dispersion regimes. From a detailed linear stability analysis, we find that the atomic detuning frequency has a strong influence on the MI. That is, the atomic detuning frequency induces nonconventional MI sidebands at the photonic band gap (PBG) edges and near the PBG edges. Especially in the normal dispersion regime, MI occurs without any threshold condition, which is in contrast with that of conventional fiber Bragg gratings. We also perform a numerical analysis to solve the NLCM-MB equations. The numerical results of the prediction of both the optimum modulation wave number and the optimum gain agree well with that of the linear stability analysis. Another main result of the present work is the prediction of the existence of both bright and dark self-induced transparency gap solitons at the PBG edges.
A two-level medium, described by the Maxwell-Bloch system, is engraved by establishing a standing cavity wave with a linearly polarized electromagnetic field that drives the medium on both ends. A light pulse, polarized along the other direction, then scatters the medium and couples to the cavity standing wave by means of the population inversion density variations. We demonstrate that control of the applied amplitudes of the grating field allows one to stop the light pulse and to make it move backward (eventually to drive it freely). A simplified limit model of the Maxwell-Bloch system with variable boundary driving is obtained as a discrete nonlinear Schrödinger equation with tunable external potential. It reproduces qualitatively the dynamics of the driven light pulse.
Abstract This paper describes the dynamics of solitonic pulse propagation in a closed loop quantum system comprised of five atomic energy levels. In this system four higher energy levels with four control laser fields form a closed ring-shape loop which is coherently coupled to a ground level state by a tunable probe laser field. Although the propagation of an intense probe pulse suffers losses and broadening in linear regime, there exist specific conditions where the probe pulse is found to become robust during its propagation. We attribute this regime to the formation of an optical soliton with a slow group velocity because of the balance between the dispersion and Kerr optical nonlinearity. Results obtained are based on the theoretical model using the coupled Maxwell–Bloch equations for the nonlinear propagation.
We theoretically show that a self-induced transparency (SIT) soliton and a Bragg soliton can coexist in a nonlinear photonic band gap (PBG) medium doped uniformly with inhomogeneous-broadening two-level atoms. The Maxwell-Bloch equations for the pulse propagating through such a uniformly doped PBG structure are derived first and further reduced to an effective nonlinear Schrödinger equation. This model describes an equivalent physical mechanism for a Bragg-soliton propagation resulting from the effective quadratic dispersion balancing with the effective third-order nonlinearity. Because the resonant atoms are taken into account, the original band gap can be shifted both by the dopants and the instantaneous nonlinearity response originating from an intense optical pulse. As a result, even if a SIT soliton with its central frequency deep inside the original forbidden band, it still can propagate through the resonant PBG medium as long as this SIT soliton satisfies the effective Bragg-soliton propagation. An approximate soliton solution describing such coexistence is found. We also show that the pulse width and group velocity of this soliton solution can be uniquely determined for given material parameters, atomic transition frequency, and input central frequency of the soliton. The numerical examples of the SIT soliton in a one-dimensional As2S3-based PBG structure doped uniformly with Lorentzian line-shape resonant atoms are shown. It is found that a SIT soliton with approximately 100-ps width in such a resonant PBG structure can travel with the velocity being two orders of magnitude slower than the light speed in an unprocessed host medium.
We consider the problem of the propagation of short-duration light pulses in a two-level medium in the regime of strong interaction, near the resonance, with no restriction on the strength of the coupling between field and medium. We prove that the local variations of the population difference created by the electric field actually induce a nonlinear effect by coupling the fundamental of the field Fourier component to its harmonics. As a result, the multiscale averaging limit of the Maxwell-Bloch system is a new system of coupled nonlinear Schrodinger equations for the slowly varying envelope of the electric field. This system has one integrable limit when the electric field is circularly polarized: it is the scalar nonlinear Schrodinger equation. Moreover, it possesses three constants of the motion and a Hamiltonian formulation. Then we prove that our system is nonintegrable as it does not pass the Painleve test. Finally, a stability analysis shows that it is modulationally unstable and thus it propagates steady pulses of coherent light.