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Physical Review A

Analytic solutions of Bloch and Maxwell-Bloch equations in the case of arbitrary field amplitude and phase modulation

А. В. Алексеев, N. V. Sushilov

Published Jul 1, 1992• 17 citations
Publisher

Abstract

The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.

DOI
10.1103/physreva.46.351

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The matrix exponent method which we applied recently for the solution of the Bloch equations [Sov. Phys. JETP 62, 1125 (1985)] allows us to obtain in the present paper the Bloch-equation solutions when the amplitude and the phase of the exciting field are arbitrary functions of time. The solution's validity conditions are derived. It is shown that one can also apply the matrix exponent method to the solution of the Maxwell-Bloch equations. The equation describing the evolution of the area of a pulse with modulated envelope and phase is obtained. This equation has an analytic solution which is a generalization of the well-known McCall-Hahn solution.
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An accurate and effective numerical method is presented to solve the Maxwell-Bloch equations,which describe the optical pulse propagation and interaction with homogeneously broadened two-level medium.The convergence and stability of this numerical method are also proved by comparing with the analytic solutions derived under special condition,and the method maintains its erroneous exponents and is applicable.Simulations of arbitrary conditions are discussed by employing this new method,and the evolutions of pulse and Bloch vectors for different detunings,relaxation times and initial input pulse power are analyzed.The established numerical method can be used to solve the Maxwell-Bloch equations and their corrections.
Coherent propagation of an optical pulse in a medium with a second-order resonance is investigated when we include both the dynamical Stark effect and a population-dependent refraction index. We show how to obtain the general solution when the incident pulse has no phase-modulation. The amplitude-modulation of the incident pulse can be arbitrary as can be the initial state of the medium. This general solution is reduced to obtaining the solution of an ordinary differential equation and quadratures. For an initially de-excited medium, the solution reduces to that given by Poluektov et al. {I. A. Poluektov, Yu. M. Popov and V. S. Roitberg, 1975, Kvant. Elektron. (Moscow), 2, 1147 [1975, Sov. J. Quant. Electron., 5, 620]}. We give some illustrative examples of solutions for various initial profiles. It is also shown that the oscillating two-stream instability (OTSI) from plasma physics is equivalent to the Maxwell-Bloch equations for degenerate two-photon propagation. We give a solution for the OTSI which contains five arbitrary real functions.
An analytic solution of the Bloch equation is presented using a class of asymmetric complex functions as a driving function. These may involve both amplitude and frequency modulation functions. Analytic solutions that have been previously derived include the amplitude modulated (``real'') hyperbolic secant pulse and its asymmetric generalization in addition to the complex sech/tanh pulse. The solution presented in this paper contains all of these as particular cases. The solution in its complex form is closely associated with an adiabatic fast passage experiment and renders robust magnetization inversion provided the rf amplitude exceeds a certain threshold. \textcopyright{} 1996 The American Physical Society.
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).
The complete solution of the Maxwell-Bloch equations by the inverse scattering transform as given by Ablowitz, Kaup, and Newell is compared with the McCall-Hahn theory, and others. Not only it is shown that the earlier results are contained in the complete solution, but also several new results are obtained. Among these are an infinite set of "nonlinear moments" which evolve similar to the McCall-Hahn area, a closed-form solution for the nonlinear transmission, how one can determine absolute time delays, why the threshold area for lossless propagation is still exactly $\ensuremath{\pi}$ (even when the initial profile is off resonance but unchirped), and first-order effects of relaxation on $2\ensuremath{\pi}$ hyperbolic-secant pulse propagation. For the last-mentioned results, we find that in general if a $2\ensuremath{\pi}$ hyperbolic-secand pulse is initially off resonance, it will move away from resonance. Also, we find that the McCall-Hahn result for first-order effects on the time delay must be modified for small-amplitude $2\ensuremath{\pi}$ hyperbolic-secant pulses.