Oscillatory Behavior of Large Eigenvalues in Quantum Rabi Models

Monvel, Anne Boutet de; Zielinski, Lech

doi:10.1093/imrn/rny294

Cited by 14 publications

(16 citation statements)

References 18 publications

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“…Since then, various aspect of the QRM and its generalizations have been discussed widely and intensively, and developed from the theoretical viewpoint (see [68] and references therein). For instance, for large eigenvalues of the quantum Rabi model a three-term asymptotic formula with an oscillatory term was recently obtained in [12].…”

Section: Introduction and Overviewmentioning

confidence: 99%

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

Kimoto

Reyes-Bustos

Wakayama

2020

International Mathematics Research Notices

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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 ε Rabi of the AQRM is defined by adding the fluctuation term εσ x , with σ x being the Pauli matrix, to the Hamiltonian of the quantum Rabi model, breaking its Z 2 -symmetry. The spectrum of H ε Rabi 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 ε 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 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.

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Section: Introduction and Overviewmentioning

confidence: 99%

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

Kimoto

Reyes-Bustos

Wakayama

2020

International Mathematics Research Notices

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“…In [27] the energy level crossing of spectral curves of asymmetric quantum Rabi model with ǫ = 1/2 is investigated, and then it is studied in [35,33,25] from mathematics point of view. In [3] the oscillation of map n → E n (g) is explicitly given for fixed g. While we are interested in the asymptotic behavior of E n (g) + g 2 as |g| → ∞. Numerically it is expected as…”

Section: Renormalized Spectral Zeta Function Of Quantum Rabi Modelmentioning

confidence: 99%

Ground states and associated path measures in the renormalized Nelson model

Hiroshima

Matte

2021

Rev. Math. Phys.

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We prove the existence, uniqueness, and strict positivity of ground states of the possibly massless renormalized Nelson operator under an infrared regularity condition and for Kato decomposable electrostatic potentials fulfilling a binding condition. If the infrared regularity condition is violated, then we show non-existence of ground states of the massless renormalized Nelson operator with an arbitrary Kato decomposable potential. Furthermore, we prove the existence, uniqueness, and strict positivity of ground states of the massless renormalized Nelson operator in a non-Fock representation where the infrared condition is unnecessary. Exponential and superexponential estimates on the pointwise spatial decay and the decay with respect to the boson number for elements of spectral subspaces below localization thresholds are provided. Moreover, some continuity properties of ground state eigenvectors are discussed. Byproducts of our analysis are a hypercontractivity bound for the semigroup and a new remark on Nelson’s operator theoretic renormalization procedure. Finally, we construct path measures associated with ground states of the renormalized Nelson operator. Their analysis entails improved boson number decay estimates for ground state eigenvectors, as well as upper and lower bounds on the Gaussian localization with respect to the field variables in the ground state. As our results on uniqueness, positivity, and path measures exploit the ergodicity of the semigroup, we restrict our attention to one matter particle. All results are non-perturbative.

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“…One of the motivations is the potential application of the QRM to the development quantum computation and quantum information sciences. In parallel, there is a growing interest in the mathematical study of the properties of the QRM, including the study of solutions and dynamics [5,6], large spectral asymptotics [7], symmetry and degeneracy for asymmetric QRMs [8,9], entanglement properties [10], Floquet analysis [11], spectral zeta functions [12,13] and algebro-geometric analysis [14,15].…”

Section: Introductionmentioning

confidence: 99%

The heat kernel of the asymmetric quantum Rabi model

Reyes-Bustos

2023

J. Phys. A: Math. Theor.

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In this paper we derive an explicit formula for the heat kernel of the asymmetric quantum Rabi model (AQRM), a symmetry breaking generalization of the quantum Rabi model (QRM). The method described here is the extension of a recently developed method for the heat kernel of the QRM that uses the Trotter-Kato product formula instead of path integrals or stochastic methods. In addition to the heat kernel formula, we give applications including the explicit formula for the partition function and the Weyl law for the distribution of the eigenvalues, obtained from the corresponding spectral zeta function.

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Oscillatory Behavior of Large Eigenvalues in Quantum Rabi Models

Cited by 14 publications

References 18 publications

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

Determinant Expressions of Constraint Polynomials and the Spectrum of the Asymmetric Quantum Rabi Model

Ground states and associated path measures in the renormalized Nelson model

The heat kernel of the asymmetric quantum Rabi model

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