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Physical Review A

Two-level systems driven by modulated pulses

Edward J. Robinson

Published Jun 1, 1985• 18 citations
Publisher

Abstract

This paper generalizes some recent results of F. T. Hioe [Phys. Rev. A 30, 2100 (1984)]. It is found that a two-level system driven by a resonant pulse with temporally symmetric amplitude modulation and antisymmetric frequency modulation may be completely inverted. In addition, it is shown that certain amplitude-modulated-- frequency-modulated combinations may be mapped into Hioe's (hyperbolic cosecant and hyperbolic tangent) solution.

DOI
10.1103/physreva.31.3986

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This paper generalizes some recent results of F. T. Hioe [Phys. Rev. A 30, 2100 (1984)]. It is found that a two-level system driven by a resonant pulse with temporally symmetric amplitude modulation and antisymmetric frequency modulation may be completely inverted. In addition, it is shown that certain amplitude-modulated-- frequency-modulated combinations may be mapped into Hioe's (hyperbolic cosecant and hyperbolic tangent) solution.
Abstract We investigate the dynamics of a two-level quantum system driven by a laser pulse characterized by Lorentzian frequency and sub-Lorentzian amplitude modulations. Complete analytical solutions, expressed via confluent Heun functions, are derived. Our analysis reveals that explicit exact analytical solutions exist under infinite sets of specific parameter conditions. The effects of modulation parameters and initial conditions on the final transition probabilities are examined analytically and numerically. Furthermore, the method is demonstrated to be directly applicable to two closely related models with Lorentzian pulses.
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.
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.
We propose a simple method of combined synchronous modulations to generate the analytically exact solutions for a parity-time symmetric two-level system. Such exact solutions are expressible in terms of simple elementary functions and helpful for illuminating some generalizations of appealing concepts originating in the Hermitian system. Some intriguing physical phenomena, such as stabilization of a non-Hermitian system by periodic driving, non-Hermitian analogs of coherent destruction of tunneling, and complete population inversion (CPI), are demonstrated analytically and confirmed numerically. In addition, by using these exact solutions we derive a pulse area theorem for such non-Hermitian CPI in the parity-time symmetric two-level system. Our results may provide an additional possibility for pulse manipulation and coherent control of the parity-time symmetric two-level system.
In a recent paper, Bambini and Berman [A. Bambini and P. R. Berman, Phys. Rev. A 23, 2496 (1981)] presented analytic solutions to a certain family of coherent-coupling pulses for a two-level system. They show, for nonresonant temporally asymmetric members of the class, that there are no solutions corresponding to vanishing transition probabilities. In this Comment, we examine the problem in greater generality and demonstrate that this property is the norm for asymmetric pulses, and that a vanishing transition probability is possible only if severely overdetermined conditions are satisfied.