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Journal of Physics B Atomic and Molecular Physics

A general solution to the two-level problem: further results

Edward J. Robinson

Published Sep 28, 1985• 10 citations
Publisher

Abstract

The author extends previous work on the eigenvalue (EV) approach to the determination of transition probabilities in two-level systems. The new results make use of the inverting EV, A i 2 , which are related to interaction strengths which completely transfer the population from the initially occupied state (level 1) to the originally empty state (level 2), as well as the restoring EV, A r 2 .

DOI
10.1088/0022-3700/18/18/004

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Physical Review A•May 13, 2002•3 citations
The author extends previous work on the eigenvalue (EV) approach to the determination of transition probabilities in two-level systems. The new results make use of the inverting EV, A i 2 , which are related to interaction strengths which completely transfer the population from the initially occupied state (level 1) to the originally empty state (level 2), as well as the restoring EV, A r 2 .
This paper analyzes the dynamics of two-level systems from a new perspective. The notion of eigenvalues (EV) of the pulse area squared, which was introduced in a previous paper, is developed further. Integral expressions for the eigenvalues in terms of eigenfunctions are exhibited, and it is pointed out that these formulas also enable one to estimate EV variationally. Examples are given of how the method may be used to predict qualitative features of two-level spectra without solving equations of motion. Finally, an expression for the transition amplitude in terms of an eigenfunction expansion is derived. It is suggested that this expansion may be useful for both numerical determinations of transition probabilities and for analytic, variational calculations of these parameters. As an application, an expression for the transition probability at small detunings for a rather general class of coupling functions is derived, and its predictions compared to exact numerical calculations in a specific case.
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 find the exact solution to the three-state problem for a class of intuitive and counterintuitive sequences of delayed pulses of finite duration in terms of the Clausen's generalized hypergeometric function, which reduces to simple analytic expressions, involving elementary functions only, for final occupation probabilities. These analytic results show that the sequence of delayed pulses, independently of the pulse order and applied detunings, can completely remove the population from the initially populated state (thus creating a quantum superposition of two other involved states). This conclusion extends the original result of Vitanov and Stenholm [Phys. Rev. A 55, 648 (1997)], to the case of nonzero two-photon detuning and more general pulse shapes.