Inverse scattering transform for two-level systems with nonzero background

Abstract

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 dominant, i.e., initial population inversion is negative. arXiv:1907.06231v1 [nlin.SI] 14 Jul 2019 their parameters and inputs, the two-level MBEs exhibit a rich variety of dynamical regimes. While some regimes of laser operation are chaotic [29][30][31][32][33], and can be described by the Lorenz equations [34,35], or are even turbulent [36][37][38][39], and have a finite-dimensional attractor [40][41][42][43] or a slow manifold [44], other regimes of light propagating through rarefied gases may be approximated by completely-integrable, soliton-type equations [4,9,10,17,[45][46][47][48][49][50][51][52].The integrable MBEs are derived under the above-mentioned approximations, and have addressed phenomena including self-induced transparency [9, 10, 45], area theorem [46], photon echo [48], amplification [47,49,50], and superfluorescence [17,51,52]. Typically, these phenomena are addressed from the viewpoint of the initial-boundary-value or signaling problem for a finitely or infinitely long narrow tube containing the active medium, into which a narrow optical pulse is injected at one end, and whose initial state is given far in the past. The MBEs can be represented in terms of a Lax pair [45,52], thus, the initial-value problem can be solved using the inverse-scattering transform (IST) [45,[52][53][54]. The temporal piece of the Lax pair is in the AKNS form [55,56], so just as for the nonlinear Schrödinger equation appearing in [57,58], which stretches from the infinite past to the infinite future, and pulses are evolved along the spatial variable, describing its propagation from the location, where the pulse is launched into the medium. The evolution operator in the Lax pair of the MBEs has an unusual form containing a Hilbert transform, which complicates the derivation and form of the equations describing the evolution of the spectral data, as well as reflects the fact that the phenomena described by the MBEs are irrev...

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