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Journal of Physics B Atomic Molecular and Optical Physics

Asymmetrized Rosen-Zener model

Nikolay V. Vitanov

Published Apr 14, 1994• 23 citations
Publisher

Abstract

A new exactly solvable two-state model with an asymmetric pulse shape and a constant detuning is presented. It is an asymmetrized generalization of the Rosen-Zener model. The final populations of the states are expressed in terms of elementary functions and the phase of the gamma function. In contrast to the temporary asymmetric pulses studied so far, the transition probability in this model can vanish for a number of combinations of the free parameters.

DOI
10.1088/0953-4075/27/7/010

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A new exactly solvable two-state model with an asymmetric pulse shape and a constant detuning is presented. It is an asymmetrized generalization of the Rosen-Zener model. The final populations of the states are expressed in terms of elementary functions and the phase of the gamma function. In contrast to the temporary asymmetric pulses studied so far, the transition probability in this model can vanish for a number of combinations of the free parameters.
We present an analytically exactly solvable two-state quantum model, in which the coupling has a hyperbolic-tangent temporal shape and the frequency detuning is constant. The exact solution is expressed in terms of associated Legendre functions. An interesting feature of this model is that the excitation probability does not vanish, except for zero pulse area or zero detuning; this feature is attributed to the asymmetric pulse shape. Two limiting cases are considered. When the coupling rises very slowly, it is nearly linear and the tanh model reduces to the shark model introduced earlier. When the coupling rises very quickly, the tanh model reduces to the Rabi model, which assumes a rectangular pulse shape and hence a sudden switch on. Because of its practical significance, we have elaborated the asymptotics of the solution in the Rabi limit, and we have derived the next terms in the asymptotic expansion, which deliver the corrections to the amplitude and the phase of the Rabi oscillations due to the finite rise time of the coupling.
The dynamics of the Rosen–Zener transition (RZT) in interacting two-level systems within nonreciprocal systems is examined, where nonreciprocity stems from asymmetric couplings between energy levels. For the linear case, an analytical expression for the RZT probability is derived. In the nonlinear scenario, numerical experiments are utilized to investigate the impact of various initial states and scanning periods on transition probabilities under adiabatic conditions . Additionally, an equivalent classical Hamiltonian is formulated to elucidate the origin of transition asymmetry under different initial state conditions. The influence of energy level detuning on the RZT is explored, leading to the suggestion of new applications for studying parameters of two-level systems related to nonlinearity and nonreciprocity. This opens avenues for potential Rosen–Zener–Stückelberg interferometry implementations.
Abstract We present a novel time-dependent two-state model that describes a constant-amplitude level-crossing field configuration, where the frequency detuning varies within a finite interval. A distinctive feature of this configuration is that the resonance crossing always occurs asymmetrically in time, making it an asymmetric version of the second Demkov-Kunike model. The general solution of the problem is expressed in terms of two independent irreducible linear combinations of the Gauss hypergeometric functions. We analyze the asymptotes of the solution in terms of corresponding quasi-energies and calculate the final transition probability in the case when the system starts from the first quasi-energy state.
We derive 15 classes of time-dependent two-state models solvable in terms of the confluent Heun functions. These classes extend over all the known families of three- and two-parametric models solvable in terms of the hypergeometric and the confluent hypergeometric functions to more general four-parametric classes involving three-parametric detuning modulation functions. In the case of constant detuning the field configurations describe excitations of two-state quantum systems by symmetric or asymmetric pulses of controllable width and edge-steepness. The classes that provide constant detuning pulses of finite area are identified and the factors controlling the corresponding pulse shapes are discussed. The positions and the heights of the peaks are mostly defined by two of the three parameters of the detuning modulation function, while the pulse width is mainly controlled by the third one, the constant term. The classes suggest numerous symmetric and asymmetric chirped pulses and a variety of models with two crossings of the frequency resonance. 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. We derive closed form solutions for particular curves in the 3D space of the involved parameters which compose the complete return spectrum of the considered two-state quantum system.
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.