Complete-Return Spectrum for a Generalized Rosen–Zener Two-State Term-Crossing Model

Abstract

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 U0n = U0n(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 δ0the curve in {U0n,δ1} plane, i.e., the δ0= const section of the general surface is accurately approximated by an ellipse which crosses the U0n-axis at the points ±n and δ1-axis at the points δ11and 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.