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arXiv:1906.11499 · New Journal of Physics

Analytical results for the dynamics of parabolic level-crossing model

Chon‐Fai Kam, Yang Chen

Published Jan 21, 2020• 6 citations
arXivHTMLPDF

Abstract

Abstract We study the dynamics of a two-level crossing model with a parabolic separation of the diabatic energies. The solutions are expressed in terms of the tri-confluent Heun equations—the generalization of the confluent hypergeometric equations. We obtain analytical approximations for the state populations in terms of Airy and Bessel functions. Applicable expressions are derived for a large part of the parameter space. We also provide simple formulas which connect local solution in different time regimes. The validity of the analytical approximations is shown by comparing them to numerical simulations.

DOI
10.1088/1367-2630/ab6e4a

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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.
We introduce a new exactly integrable level-crossing model of quantum semiclassical two-state problem for which the analytic solution is written in terms of the Kummer confluent hypergeometric functions. This is a constant-amplitude field-configuration describing an asymmetricin- time level-crossing process for which the laser field frequency detuning is given in terms of the Lambert-W function. The general solution of the problem for this model is written as a linear combination, with arbitrary constant coefficients, of two fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. We present the fundamental solutions and analyze the behavior of the system in the external field defined by the specific field configuration.
We theoretically investigate the exact solutions for generalized parity–time( <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T )-reversal-symmetric Rabi models driven by external fields with monochromatic periodic, linear, and parabolic forms, respectively. The corresponding exact solutions are presented in terms of the confluent Heun equations without any approximation. In principle, the analytic solutions derived here are valid in the whole parameter space. Such a kind of study may offer potential coherent control schemes of the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T -symmetric two-level systems.