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Analytical eigenstates for the quantum Rabi model

Chaohong Lee

Published Aug 15, 2016• 115 citations
OpenAlex

Abstract

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.

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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.
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 have studied the two-qubit quantum Rabi model in the asymmetric case and its generalizations with dipole and Heisenberg-type qubit-qubit interactions. The solutions are obtained analytically with eigenstates given in terms of the extended coherent states or photon number states. For identical qubit-photon couplings, a novel type of quasi-exact solution which exists for all coupling values with constant eigenenergy is found, leading to level crossings within the same parity subspace even for non-identical qubits. In contrast to the quasi-exact eigenstates of the single-qubit model, these exceptional eigenstates are formed by just a few Fock states (photon number bounded from above at one or three), and the condition for them depends only on a fine-tuning of the qubit level splittings with respect to the photon energy, but not on the coupling to the photon field if the qubit-qubit interaction is not taken into consideration. This makes them excellent candidates for direct experimental observation within circuit quantum electrodynamics (QED) and application in single photon experiments. Besides, exceptional solutions with finite photon numbers N are also found.
Within extended coherent states, a recent exact solution to the quantum Rabi model (QRM) [D. Braak, Phys. Rev. Lett. 107, 100401 (2011)] can be recovered in an alternative simpler and more physical way, without use of any extra conditions. In the same framework, the two-photon QRM is solved exactly by treating extended squeezed states on an equal footing. Concise transcendental functions responsible for the exact solutions are derived. The isolated Juddian solutions are also analytically obtained in terms of degeneracy. Both the extended coherent states and squeezed states employed here are essentially Fock states in the space of the corresponding Bogoliubov operators, which result in free-particle number operators. The present approach can be summarized concisely in a unified way and easily extended to various spin-boson systems with multiple levels, even multiple modes.
This paper is concerned with the rigorous analytical determination of the spectrum of the two-photon and the two-mode quantum Rabi models. To reach this goal, we exploit the hidden symmetries in these models by means of the unitary and similarity transformations in addition to the Bargmann-Fock space description. In each case, the purely quantum mechanical problem of the Rabi model studied is reduced to solutions for differential equations. This eventually gives a third-order differential equation for each of these models, which is reduced to a second-order differential equation by additional transformations. The analytical expressions of the wave functions describing the energy levels are obtained in terms of the confluent hypergeometric functions.