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arXiv:2405.10117 · Studies in Applied Mathematics

On the Coupled Maxwell–Bloch System of Equations With Nondecaying Fields at Infinity

Sitai Li, Gino Biondini, Gregor Kovačič

Published May 1, 2025• 1 citation
arXivHTMLPDF

Abstract

ABSTRACT We study an initial‐boundary‐value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two‐level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio‐temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse.

DOI
10.1111/sapm.70055

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ABSTRACT We study an initial‐boundary‐value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two‐level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio‐temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse.
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
In this paper, a study of the coupled Schrödinger Maxwell‐Bloch equations, which described the propagation of two optical pulses in an optical medium with coherent three‐level atoms, is presented via the Riemann‐Hilbert approach. First, we performed a spectral analysis on the basis of Lax pair with the negative flow, and then the Jost function, scattering matrix, and their analytic and symmetric properties are given. Second, the Riemann‐Hilbert problem is established successfully through a standard dressing procedure via the Riemann‐Hilbert approach, and then the potential function related to the solution of the Riemann‐Hilbert problem is reconstructed. By introducing the special matrix functions, we can transform the irregular Riemann‐Hilbert problem into the regular one, which can be solved by the Plemelj formula. Finally, some applications are given to solve the Riemann‐Hilbert problem without reflection for the coupled Schrödinger Maxwell‐Bloch equation, and the multi‐soliton solutions are obtained explicitly. Moreover, the dynamic behaviors of some soliton solutions are discussed graphically by choosing appropriate parameters.
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.
We extend to more than one spatial dimension the semiclassical full-wave vector Maxwell-Bloch equations for the purpose of achieving an adequate and rigorous description of ultrashort pulse propagation in optical waveguides containing resonant nonlinearities. Our considerations are based on the generalized pseudospin formalism introduced by Hioe and Eberly [Phys. Rev. Lett. 47, 838 (1981)] for treatment of the resonant coherent interactions of ultrashort light pulses with discrete-multilevel systems. A self-consistent set of coupled curl Maxwell-pseudospin equations in two spatial dimensions and time for the special case of a degenerate three-level system of quantum absorbers is originally derived. Maxwell's curl equations are considered to be coupled via macroscopic medium polarization to the three-level atom model for the resonant medium. Two distinct sets of pseudospin equations are obtained corresponding to the TE- and TM-polarized optical waves. For the case of TM polarization, the electromagnetic wave is polarized in a general direction in the plane of incidence inducing two dipole transitions in a degenerate three-level system by each E-field component along the propagation axis and in transverse direction. We introduce a dipole-coupling interaction Hamiltonian allowing Rabi flopping of the population difference along and perpendicular to the propagation axis with frequencies depending on the corresponding field components. The relationship between the induced polarization and the state vector components that describe the evolution of the discrete-level system is derived in order to couple the quantum system equations to the Maxwell's curl equations. The pseudospin equations are phenomenologically extended to include relaxation effects by introducing nonuniform decay times corresponding to the various dipole transitions occurring in a three-level system. The system has been discretized using finite differences on a Yee grid and solved numerically by an iterative predictor-corrector finite-difference time-domain method. Self-induced transparency soliton propagation through a degenerate three-level quantum system of absorbers in two spatial dimensions and time is demonstrated in planar parallel-mirror waveguide geometries.
We formulate the inverse scattering transform for the scalar Maxwell-Bloch system of equations describing the resonant interaction of light and active optical media in the case when the light intensity does not vanish at infinity. We show that pure background states in general do not exist with a nonzero background field. We then use the formalism to compute explicitly the soliton solutions of this system. We discuss the initial population of atoms and show that the pure soliton solutions do not correspond to a pure state initially. We obtain a representation for the soliton solutions in determinant form and explicitly write down the one-soliton solutions. We next derive periodic solutions and rational solutions from the one-soliton solutions. We then analyze the properties of these solutions, including discussion of the sharp-line and small-amplitude limits, and thereafter show that the two limits do not commute. Finally, we investigate the behavior of general solutions, showing that solutions are stable (i.e., the radiative parts of solutions decay) only when initially atoms in the ground state dominate, i.e., initial population inversion is negative.
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).