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Chinese Physics B

Analytic solutions for generalized PT -symmetric Rabi models*

Yuanhao Dong, Wenjing Zhang, Jing Liu, Xiao-Tao Xie

Published Sep 18, 2019• 4 citations
Publisher

Abstract

We theoretically investigate the exact solutions for generalized parity–time( <?CDATA ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> P T -symmetric two-level systems.

DOI
10.1088/1674-1056/ab457b

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We theoretically investigate the exact solutions for generalized parity–time( <?CDATA ${\mathscr{P}}{\mathscr{T}}$?> 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 ${\mathscr{P}}{\mathscr{T}}$?> P T -symmetric two-level systems.
We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak [Phys. Rev. Lett. \textbf{107}, 100401 (2011)] are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.
We develop a method to find analytical solutions for the eigenstates of the quantum Rabi model. These include symmetric, anti-symmetric and asymmetric analytic solutions given in terms of the confluent Heun functions. Both regular and exceptional solutions are given in a unified form. In addition, the analytic conditions for determining the energy spectrum are obtained. Our results show that conditions proposed by Braak (2011 Phys. Rev. Lett. 107 100401) are a type of sufficiency condition for determining the regular solutions. The well-known Judd isolated exact solutions appear naturally as truncations of the confluent Heun functions.
It is shown that a semiclassical Rabi model with parity-time ( PT ) symmetry possesses quasiexact solutions, which can be found using the Lie-algebraic approach. These solutions are located precisely at the exceptional points of the spectrum, the boundaries of the PT -symmetric phase. The corresponding constraints on the model parameters can be interpreted as a resonance relationship between the constant and periodic driving terms. For sufficiently large resonance ratio and judiciously chosen parameters, we find that the time evolution of the complete qubit can be described exactly.
We theoretically investigate the dipolar molecular system driven by monochromatic periodic, linear, parabolic, and sech2{\mathrm{sech}}^{2}sech2 forms external fields, respectively. The two-level Hamiltonian model with nonzero diagonal dipole matrix elements is adopted to describe the population dynamics of the driven dipolar molecule, and the corresponding exact solutions are presented in terms of the confluent Heun equations without the generalized rotating-wave approximation. The analytic solutions derived here are valid in the whole parameter space.
We propose a simple method of combined synchronous modulations to generate the analytically exact solutions for a parity-time symmetric two-level system. Such exact solutions are expressible in terms of simple elementary functions and helpful for illuminating some generalizations of appealing concepts originating in the Hermitian system. Some intriguing physical phenomena, such as stabilization of a non-Hermitian system by periodic driving, non-Hermitian analogs of coherent destruction of tunneling, and complete population inversion (CPI), are demonstrated analytically and confirmed numerically. In addition, by using these exact solutions we derive a pulse area theorem for such non-Hermitian CPI in the parity-time symmetric two-level system. Our results may provide an additional possibility for pulse manipulation and coherent control of the parity-time symmetric two-level system.