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Physical Review A

Solution of Bloch equations involving amplitude and frequency modulations

F. T. Hioe

Published Oct 1, 1984• 191 citations
Publisher

Abstract

The undamped Bloch equations are shown to have a more general analytic solution which includes the famous self-induced-transparency solution of McCall and Hahn, and the nonadiabatic excitation solution of Allen and Eberly as special cases. A generalized area characteristic for a laser pulse of modulated amplitude and frequency of the forms of hyperbolic secant and tangent is given. It resolves the difficulty of the previous area concept in the presence of frequency modulation. The solution provides an exact analytic result for the amplitude- and frequency-modulated pulse of any area.

DOI
10.1103/physreva.30.2100

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The undamped Bloch equations are shown to have a more general analytic solution which includes the famous self-induced-transparency solution of McCall and Hahn, and the nonadiabatic excitation solution of Allen and Eberly as special cases. A generalized area characteristic for a laser pulse of modulated amplitude and frequency of the forms of hyperbolic secant and tangent is given. It resolves the difficulty of the previous area concept in the presence of frequency modulation. The solution provides an exact analytic result for the amplitude- and frequency-modulated pulse of any area.
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.
We have found new analytic results for a wide variety of atomic excitation processes which involve the use of amplitude-and frequency-modulated laser pulses: (1) We have unified all the previously known solutions including those of Rosen and Zener,1 Landau2 and Zener,3 Rabi,4 Bambini and Berman,5 Demkov,6 McCall and Hahn,7 Allen and Eberly,8 and Hioe,9 among others, and have significantly extended it with expressions involving three free parameters and an arbitrary function to an infinite variety of amplitude-and frequency-modulated laser pulses.10 We have found a simple condition which the three essential parameters in the amplitude-and frequency-modulation functions must satisfy in order to achieve a complete population inversion, or to achieve a complete population restoration as in the case of self-induced transparency7 or soliton propagation.
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.
Analytic solutions for a class of pulses that includes an infinite variety of amplitude and frequency modulations which we previously obtained are applied specifically to the square, Gaussian, Lorentzian, hyperbolic-secant, and the exponential pulses, among others. For all the different types of amplitude- and its corresponding frequency-modulated functions, the outcome of the atomic excitation is given by the same simple analytic formulas. Our analytic solutions apply equally to two-state collision problems.
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.
A more accurate description of the 2 pi pulse of self-induced transparency is obtained. This is an exact solution of an approximate form of the Maxwell-Bloch equations in which only backscattering is neglected; the equations are valid at densities less than about 10 18 atoms cm -3 . At such densities the 2 pi sech pulse of McCall and Hahn (1967, 1969) remains a good approximate well into the picosecond range. The more accurate solution is chirped by a factor proportional to the square of the ratio of the spectral width of the pulse to its carrier wave frequency. The stability of this pulse solution and other N soliton solutions of the approximate Maxwell-Bloch equations is demonstrated.