2013, Vol.16, No.1, pp.86-92

The general semiclassical time-dependent two-state problem is considered for a specific field configuration referred to as the generalized Rosen-Zener model. This is a rich family of pulse amplitude- and phase-modulation functions describing both non-crossing and term-crossing models with one or two crossing points. The model includes the original constant-detuning non-crossing Rosen-Zener model as a particular case. We show that the governing system of equations is reduced to a confluent Heun equation. When inspecting the conditions for returning the system to the initial state at the end of the interaction with the field, we reformulate the problem as an eigenvalue problem for the peak Rabi frequency and apply the Rayleigh-Schrödinger perturbation theory. Further, we develop a generalized approach for finding the higher-order approximations, which is applicable for the whole variation region of the involved input parameters of the system. We examine the general surface U_0n =U_0n (δ₀ ,δ₁), n=const, in the three-dimensional space of input parameters, which defines the position of the n-th order return-resonance, and show that for fixed δ₀ the curve in {U_0n,δ₁} plane, i.e., the δ₀=const section of the general surface is accurately approximated by an ellipse which crosses the δ₀-axis at the points ±n and δ₁-axis at the points δ₁₁ and δ₁₂. We find a highly accurate analytic description of the functions δ₁₁(δ₀,n) and δ₁₂(δ₀,n) as the zeros of a Kummer confluent hypergeometric function. From the point of view of the generality, the analytical description of mentioned curve for the whole variation range of all involved parameters is the main result of the present paper.

Key words: two-state problem, Rosen - Zener model, confluent Heun equation, eigenvalue problem, Rabi frequency

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