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Physical Review A

Analytic solutions to the two-state problem for a class of coupling potentials

Arturo Bambini, P. R. Berman

Published May 1, 1981• 211 citations
Publisher

Abstract

A class of pulse functions is found for which analytic solutions to the problem of two levels coupled by these pulse functions is obtained. The hyperbolic-secant coupling pulse is included in this class of functions leading to the Rosen-Zener solution, but all other pulses belonging to the class function are asymmetric. The asymmetric pulses lead to qualitatively new features in the solutions; in general, it is impossible to have a zero-transition probability with such asymmetric pulses.

DOI
10.1103/physreva.23.2496

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In the concluding paragraph of a recent paper on two-state atoms driven by transients, Vitanov (1994) raised two questions. First, he queried whether or not, at non-zero detuning, there exist asymmetric pulses of arbitrary smoothness that can restore the system to its initial level. Second, he asked whether one could find symmetric pulses that fail to restore the system to its initial state, regardless of pulse area. In this letter, we respond to these speculations. We show that pulses with shapes that are hyperbolic secants modified by powers of the hyperbolic tangent can arbitrarily approach the restoring condition, while the existence of a symmetric pulse incapable of restoration implies unitarity violation. We also draw a distinction between the infinitely large set of restoring pulse areas that exist for symmetric pulses, as compared to the small number (typically no more than one) that can be found in the asymmetric case.

Journal of Physics B Atomic Molecular and Optical Physics•Mar 28, 1995•2 citations
A class of pulse functions is found for which analytic solutions to the problem of two levels coupled by these pulse functions is obtained. The hyperbolic-secant coupling pulse is included in this class of functions leading to the Rosen-Zener solution, but all other pulses belonging to the class function are asymmetric. The asymmetric pulses lead to qualitatively new features in the solutions; in general, it is impossible to have a zero-transition probability with such asymmetric pulses.
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 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.
In the concluding paragraph of a recent paper on two-state atoms driven by transients, Vitanov (1994) raised two questions. First, he queried whether or not, at non-zero detuning, there exist asymmetric pulses of arbitrary smoothness that can restore the system to its initial level. Second, he asked whether one could find symmetric pulses that fail to restore the system to its initial state, regardless of pulse area. In this letter, we respond to these speculations. We show that pulses with shapes that are hyperbolic secants modified by powers of the hyperbolic tangent can arbitrarily approach the restoring condition, while the existence of a symmetric pulse incapable of restoration implies unitarity violation. We also draw a distinction between the infinitely large set of restoring pulse areas that exist for symmetric pulses, as compared to the small number (typically no more than one) that can be found in the asymmetric case.