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Acta Physica Sinica

Exact solution to a kind of multi-level Rosen-Zener models

Shao-Wu Yao, Hong Cao, Li-Xiang Cen

Published Jan 1, 2019• 0 citations
Publisher

Abstract

Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.

DOI
10.7498/aps.68.20190353

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Exact solution to the driven quantum system with an explicitly time-dependent Hamiltonian is not only an issue of fundamental importance to quantum mechanics itself, but also a ubiquitous problem in the design for quantum control. In particular, the nonadiabatic transition induced by the time-dependent external field is often involved in order to target the quantum state for the atomic and molecular systems. In this paper we investigate the exact dynamics and the associated nonadiabatic transition in a typical driven model, the Rosen-Zener model and its multi-level extension, by virtue of the algebraic dynamical method. Previously, this kind of driven models, especially of the two-level case, were solved by converting the corresponding Schrödinger equation to a hypergeometric equation. The property of the dynamical transition of the system was then achieved by the asymptotic behavior of the yielded hypergeometric function. A critical drawback related to such methods is that they are very hard to be developed so as to treat the multi-level extension of the driven model. Differing from the above mentioned method, we demonstrate that the particular kind of the Rosen-Zener model introduced here could be solved analytically via a canonical transformation or a gauge transformation approach. In comparison, we show that the present method at least has two aspects of advantages. Firstly, the method enables one to describe the evolution of the wavefunction of the system analytically over any time interval of the pulse duration. Moreover, we show that the method could be exploited to deal with the multi-level extensions of the model. The explicit expression of the dynamical basis states, including the three-level system and the four-level system, is presented and the transition probabilities induced by the nonadiabatic evolution among different levels are then characterized for the model during the time evolution. In addition, our study reveals further that the dual model of the driven system can be constructed. Since the dynamical invariant of a solvable system can always be obtained within the framework of the algebraic dynamical method, the general connection between the dual model and the original one, including the solvability and their dynamical invariants, are established and characterized distinctly.
A class of exact analytic solutions of the time-dependent Schrödinger equation is presented for a two-state quantum system coherently driven by a nonresonant external field. The coupling is a linear function of time with a finite duration and the detuning is constant. Four special models are considered in detail, namely the shark, double-shark, tent and zigzag models. The exact solution is derived by rotation of the Landau–Zener propagator at an angle of π/4 and is expressed in terms of Weber's parabolic cylinder function. Approximations for the transition probabilities are derived for all four models by using the asymptotics of the Weber function; these approximations demonstrate various effects of physical interest for each model.
We investigate an explicitly time-dependent quantum system driven by a secant-pulse external field. By solving the Schr\"odinger equation exactly, we elucidate exotic properties of the system with respect to its dynamical evolution: On the one hand, the system is shown to be essentially nonadiabatic, which prohibits an adiabatic approximation for its dynamics; on the other hand, the loop evolution of the model can induce a geometric phase which, analogous to the Berry phase of the cyclic adiabatic evolution, is in direct proportion to the solid angle subtended by the path of the state vector. Moreover, we extend the model and show that the described properties coincide in a special family of secant-pulse-driven models.
We develop the group-theoretical approach for analytically solving the time-dependent Schrodinger equations for multilevel atoms driven by amplitude- and/or frequency-modulated concurrent laser fields. The time evolution of externally driven N-level atoms is calculated in an explicit matrix form. The exact analytical conditions for complete population transfer and its return are simply connected with the parameters of the symmetry. We consider in detail the case of a four-level system driven by concurrent laser pulses of different forms.
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.