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arXiv:1402.1321

Complete-return spectrum for a generalized Rosen-Zener two-state term-crossing model

T. A. Shahverdyan, Mogilevtsev, D. S., Red'kov, V. M., Artur Mikhailovich Ishkhanyan

Published Feb 6, 2014• 3 citations
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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 in the 3D space of input parameters, which defines the position of the n-th order return-resonance, and show that the section of the general surface is accurately approximated by an ellipse. We find a highly accurate analytic description through 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.

DOI
10.48550/arxiv.1402.1321

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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 in the 3D space of input parameters, which defines the position of the n-th order return-resonance, and show that the section of the general surface is accurately approximated by an ellipse. We find a highly accurate analytic description through 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.
We discuss the level-crossing field configurations for which the quantum time-dependent two-state problem is solvable in terms of the confluent Heun functions. We show that these configurations belong to fifteen four-parametric families of models that generalize all the known 3- and 2-parametric families for which the problem is solvable in terms of the Gauss hypergeometric and the Kummer confluent hypergeometric functions. Analyzing the general case of variable Rabi frequency and frequency detuning we mention that the most notable features of the models provided by the derived classes are due to the extra constant term in the detuning modulation function. Due to this term the classes suggest numerous symmetric or asymmetric chirped pulses and a variety of models with two crossings of the frequency resonance. The latter models are generated by both real and complex transformations of the independent variable. In general, the resulting detuning functions are asymmetric, the asymmetry being controlled by the parameters of the detuning modulation function. In some cases, however, the asymmetry may be additionally caused by the amplitude modulation function. We present an example of the latter possibility and additionally mention a constant amplitude model with periodically repeated resonance-crossings. Finally, we discuss the excitation of a two-level atom by a pulse of Lorentzian shape with a detuning providing one or two crossings of the resonance. Using a series expansion of the solution of the confluent Heun equation in terms of the Kummer hypergeometric functions we derive particular closed form solutions of the two-state problem for this field configuration. The particular sets of the involved parameters for which these solutions are obtained define curves in the 3D space of the involved parameters belonging to the complete return spectrum of the considered two-state quantum system.
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Abstract We present a new exactly integrable time-dependent constant-amplitude level-crossing two-state quantum model for which the detuning of the excitation laser field varies with time over a restricted interval. This field configuration is a member of one of the eleven independent classes of general Heun two-state models. We prove that this is the only non-classical unconditionally solvable field configuration among the general Heun classes, solvable in terms of finite sums of Gauss hypergeometric functions. Each of the two fundamental solutions that compose the general solution of the problem is written as an irreducible linear combination of two ordinary hypergeometric functions.
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