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arXiv:1609.00434 · Journal of Physics A Mathematical and Theoretical

The quantum Rabi model: solution and dynamics

Qiongtao Xie, Honghua Zhong, Murray T. Batchelor, Chaohong Lee

Published Feb 10, 2017• 210 citations
arXivHTMLPDF

Abstract

Abstract This article presents a review of recent developments on various aspects of the quantum Rabi model. Particular emphasis is given on the exact analytic solution obtained in terms of confluent Heun functions. The analytic solutions for various generalisations of the quantum Rabi model are also discussed. Results are also reviewed on the level statistics and the dynamics of the quantum Rabi model. The article concludes with an introductory overview of several experimental realisations of the quantum Rabi model. An outlook towards future developments is also given.

DOI
10.1088/1751-8121/aa5a65

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Abstract This article presents a review of recent developments on various aspects of the quantum Rabi model. Particular emphasis is given on the exact analytic solution obtained in terms of confluent Heun functions. The analytic solutions for various generalisations of the quantum Rabi model are also discussed. Results are also reviewed on the level statistics and the dynamics of the quantum Rabi model. The article concludes with an introductory overview of several experimental realisations of the quantum Rabi model. An outlook towards future developments is also given.
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.
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.
A general approach is proposed to the quantum Rabi model and its several variants within the extended coherent states. The solutions to all these models including the anisotropy and the nonlinear Stark coupling are then obtained in an unified way. The essential characteristics such as the possible first-order phase transition can be detected analytically. This approach can be easily applied to the recent experiments with various tunable parameters without much additional effort, so it should be very helpful to the analysis of the experimental data.
Abstract It is well known that the A-square term must be considered in both cavity and circuit quantum electrodynamics systems, because it arises in the derivation from the minimal coupling Hamiltonian at any finite coupling strength. In this paper, we study the quantum Rabi model with A-square terms using the Bogoliubov operator approach. After a unitary transformation, the A-square terms can be eliminated, resulting in a modified quantum Rabi model with renormalized parameters. A transcendental function responsible for the exact solution is then derived. The presence of the A-square terms is found to significantly alter the energy spectrum. The dynamics are also studied using the obtained exact wave function, which is sensitive to the strength of the A-square terms at strong coupling. We believe that these results could be observed in future light–matter interaction systems in the ultra-strong and deep strong coupling regimes.