Multiphoton transitions: Approximations for the effective Hamiltonian

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Abstract
The problem of calculating transition probabilities for an atom interacting with an intense radiation field is set up within the context of an effective Hamiltonian formalism. This has the virtue that resonance effects are properly accounted for at the outset; more generally, it provides a convenient framework for approximations. Through the restriction of the class of intermediate states to be included in the construction of the effective potential, a unitary (nonperturbative) model is obtained. A successive approximation procedure for determining the effective potential, which is based on continued-fraction representations, is described. While very likely intractable for the true problem, this successive approximation procedure can be implemented, for the model effective potential, with the aid of a generalized Rayleigh-Ritz procedure described here. The level-shift matrix is constructed from the effective potential; it determines the transition probabilities, linewidths, and energy-level displacements induced by the field. We derive variational expressions of the Kohn type for each of the elements of the model level-shift matrix.