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Physical Review A

Analytic solutions of the two-state problem for a class of chirped pulses

Jakub Zakrzewski

Published Dec 1, 1985• 60 citations
Publisher

Abstract

Analytic solutions to the two-state-problem Bloch equations are obtained for a class of smooth chirped pulses. The asymmetric pulses introduced by Bambini and Berman as well as the chirped hyperbolic secant pulse of Hioe belong to the class of pulses discussed.
DOI
10.1103/physreva.32.3748

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Get PDF Email Share Share with Facebook Tweet This Post on reddit Share with LinkedIn Add to CiteULike Add to Mendeley Add to BibSonomy Get Citation Copy Citation Text F. T. Hioe and C. E. Carroll, "Analytic solutions to the two-state problem for chirped pulses," J. Opt. Soc. Am. B 2, 497-502 (1985) Export Citation BibTex Endnote (RIS) HTML Plain Text Citation alert Save article

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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.
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.
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.
In this research study, we derive the exact solutions of the Bloch equations describing the dynamics of a two-level atom with dephasing. In the two-level atom, a strong laser pump couples a ground state to an upper excited state with a time-dependent Rabi-frequency. The exact solutions are given for the atomic population inversion and the real and imaginary parts of the coherence while the input pulse is an asymmetric hyperbolic cosine form. Additionally, the system is under a chirped detuning. The method of solving the Bloch equations analytically is a very tedious part of the research, and as far as we know, there are few exact solutions available in this field. Hence, our solutions might be of great interest to various research areas, including nuclear magnetic resonance, where analytical solutions to the Bloch equations play a major role in the study of the information on the state of the medium as determined by the NMR signals.