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 (19)

• Beyond the Jaynes-Cummings model: circuit QED in the ultrastrong coupling regime
• On the Integrability of the Rabi Model
• The quantum Rabi model: solution and dynamics
• Quantum Rabi Model with Trapped Ions
• Analog quantum simulation of the Rabi model in the ultra-strong coupling regime
• Semi-classical and quantum Rabi models: in celebration of 80 years
• Inversion of Qubit Energy Levels in Qubit-Oscillator Circuits in the Deep-Strong-Coupling Regime
• Full spectrum of the Rabi model
• A generalization of the quantum Rabi model: exact solution and spectral structure
• Integrability vs exact solvability in the quantum Rabi and Dicke models
• Integrability vs Quantum Thermalization
• Note on the Analytical Solution of the Rabi Model
• Symmetry of asymmetric quantum Rabi models
• Analytical energy spectrum for hybrid mechanical systems
• Addendum to `Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model'
• On the Existence of Monodromies for the Rabi model
• On a conjecture of Kimoto and Wakayama
• Asymptotic behavior of observables in the asymmetric quantum Rabi model
• New symmetry in the Rabi model

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• 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