Determinant expressions of constraint polynomials and the spectrum of the asymmetric quantum Rabi model

Abstract

The purpose of this paper is to study the exceptional eigenvalues of the asymmetric quantum Rabi models (AQRM), specifically, to determine the degeneracy of their eigenstates. Here, the Hamiltonian $H_{\text{Rabi}}^{\epsilon }$ of the AQRM is defined by adding the fluctuation term $\epsilon {\sigma }_x$, with ${\sigma }_x$ being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its ${\mathbb{Z}}_2$-symmetry. The spectrum of $H_{\text{Rabi}}^{\epsilon }$ contains a set of exceptional eigenvalues, considered to be remains of the eigenvalues of the uncoupled bosonic mode, which are further classified in two types: Juddian, associated with polynomial eigensolutions, and non-Juddian exceptional. We explicitly describe the constraint relations for allowing the model to have exceptional eigenvalues. By studying these relations we obtain the proof of the conjecture on constraint polynomials previously proposed by the third author. In fact, we prove that the spectrum of the AQRM possesses degeneracies if and only if the parameter $\epsilon$ is a half-integer. Moreover, we show that non-Juddian exceptional eigenvalues do not contribute any degeneracy and we characterize exceptional eigenvalues by representations of ${\mathfrak{sl}}_2$. Upon these results, we draw the whole picture of the spectrum of the AQRM. Furthermore, generating functions of constraint polynomials from the viewpoint of confluent Heun equations are also discussed.

68 pages. The organization of the manuscript was changed to improve clarity

References in corpus (6)

• Beyond the Jaynes-Cummings model: circuit QED in the ultrastrong coupling regime
• The quantum Rabi model: solution and dynamics
• Inversion of Qubit Energy Levels in Qubit-Oscillator Circuits in the Deep-Strong-Coupling Regime
• Integrability vs exact solvability in the quantum Rabi and Dicke models
• Note on the Analytical Solution of the Rabi Model
• Asymptotic behavior of observables in the asymmetric quantum Rabi model

Cited by in corpus (14)

• The hidden symmetry of the asymmetric quantum Rabi model
• Hidden symmetry and tunnelling dynamics in asymmetric quantum Rabi models
• Remarks on the hidden symmetry of the asymmetric quantum Rabi model
• Entanglement resonance in the asymmetric quantum Rabi model
• Double degeneracy associated with hidden symmetries in the asymmetric two-photon Rabi model
• Generalized adiabatic approximation to the asymmetric quantum Rabi model: conical intersections and geometric phases
• Spectral determinant of the two-photon quantum Rabi model
• Generalized adiabatic approximation to the quantum Rabi model
• PT-symmetric quantum Rabi model
• Non-orthogonal qubit states expansion for the asymmetric quantum Rabi model
• The heat kernel of the asymmetric quantum Rabi model
• Spectral continuum in the Rabi-Stark model
• General symmetry operators of the asymmetric quantum Rabi model
• Symmetry operators of the asymmetric two-photon quantum Rabi model