Eigenvalue problem of the square of the pulse area for two-level systems

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.