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Physical Review A

Two-level transition probabilities for asymmetric coupling pulses

Edward J. Robinson

Published Oct 1, 1981• 30 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.24.2239

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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.
It is shown that any pulse with a time-asymmetric envelope and a constant non-zero detuning excites a two-level quantum system with a probability of nearly one-half provided the pulse is very asymmetric, adiabatic in the slowly rising part of its envelope, sudden in the steep part and the peak Rabi frequency is much greater than the detuning. Three particular cases are considered analytically: the asymmetric hyperbolic-secant, the Bambini-Berman and the exponential pulse, which illustrate the general conclusions. The results are applied to a special three-level 'vee' system whose two transitions are driven by two asymmetric laser pulses of the same Rabi frequencies and the same detunings. Analysis shows that if such a system is prepared initially in its ground state, effective population depletion of this state can be achieved, the results being quite insensitive to variations in the Rabi frequency and the detuning.
The problem of calculating transition probabilities in two-level systems is studied in the limit where the detuning is large compared to the inverse duration of the interaction. Coupling potentials whose Fourier transforms $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(\ensuremath{\omega})$ are of the form $f(\ensuremath{\omega}){e}^{\ensuremath{-}(|b\ensuremath{\omega}|)}$ for large frequencies give rise to solutions which may be classified into families according to the form of $f(\ensuremath{\omega})$. Within each family transition probabilities may be calculated from formulas that differ only in the numerical value of a scaling parameter. In cases where the coupling function has a pole in the complex time plane, the families are identified with the order of this singularity. In particular, for poles of first order, a connection with the Rosen-Zener solution can be made. The analysis is performed via high-order perturbation expansions which are shown to always converge for two-level systems driven by coupling potentials of finite pulse area.
Previous results obtained for transition probabilities in two-level systems are simplified and generalised. It was shown in an earlier paper by Robinson (1984) that one could express the transition amplitude as the product of its first-order approximant and a factor which is determined by the eigenvalues and eigenfunctions of a certain nonHermitian operator. In the present work, it is demonstrated that explicit knowledge of the eigenfunctions is not needed-it is sufficient to know the ensemble of eigenvalues. As an illustration, the transition probability induces by a Lorentzian pulse slightly detuned from resonance is calculated. Unlike most pulse shapes, which exhibit power broadening, it is found that the Lorentzian is neutral to first order in the detuning, and exhibits power narrowing when quadratic terms in the detuning are included.
We have studied a special class of time-symmetric pulses interacting with a two-level system. These pulses are built out of a class of asymmetric functions for which the analytic description of the interaction was possible. The time behavior of these pulses is given. We have evaluated the transition probability in a closed form. These pulses can be used to simulate the familiar Lorentzian and Gaussian pulse shapes remarkably well. Therefore, they can be used in computing the overall features of the interaction process. Comparisons of the transition amplitudes are made between the closed-form solution and numerically calculated Lorentzian and Gaussian cases.