Inverse spectral theory for semiclassical Jaynes–Cummings systems

Floch, Yohann Le; Pelayo, Álvaro; Ngọc, San Vũ

doi:10.1007/s00208-015-1259-z

Cited by 22 publications

(31 citation statements)

References 40 publications

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“…We also remark that there has been recent interest in the integrability of the Jaynes-Cummings model and generalizations (see e.g. [39]). Moreover, we note that although there are various coupling regimes of the AQRM given in terms of ∆, ω (= 1) and g physically, the discussion in this paper is independent of the choice of regimes (cf.…”

Section: Introduction and Overviewmentioning

confidence: 89%

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: 89%

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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“…[66,46] and references therein). In [46], the authors prove the following result: Theorem 6.11 ([46]). For quantum semi-toric systems which are either semiclassical pseudo-differential operators, or semiclassical Berezin-Toeplitz operators, the joint spectrum (modulo O( 2 )) determines the following invariants:…”

Section: The General Inverse Problemmentioning

confidence: 99%

Integrable systems, symmetries, and quantization

Sepe

Ngọc

2017

Lett Math Phys

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These notes are an expanded version of a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularities and expose recent research in connection with semi-toric systems. The quantum and semiclassical counterpart are also presented, in the viewpoint of the inverse question: from the quantum mechanical spectrum, can one recover the classical system? ForewordThese notes, after a general introduction, are split into four parts:• Integrable systems and action-angle coordinates (Section 3), where the basic notions about Liouville integrable systems are recalled.• Almost-toric singular fibers (Section 4), where emphasis is laid on a Morselike theory of integrable systems.• Semi-toric systems (Section 5), where we introduce the recent semi-toric systems and their classification.

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“…Proof. We recall that the Poisson bracket is linear and that we have the identities in (19). Then we compute in the coordinates given in (2) {J,…”

Section: 4mentioning

confidence: 99%

A family of compact semitoric systems with two focus-focus singularities

Hohloch¹,

Palmer²

2018

Journal of Geometric Mechanics

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About 6 years ago, semitoric systems were classified by Pelayo & Vũ Ngo . c by means of five invariants. Standard examples are the coupled spin oscillator on S 2 × R 2 and coupled angular momenta on S 2 × S 2 , both having exactly one focus-focus singularity. But so far there were no explicit examples of systems with more than one focus-focus singularity which are semitoric in the sense of that classification. This paper introduces a 6-parameter family of integrable systems on S 2 × S 2 and proves that, for certain ranges of the parameters, it is a compact semitoric system with precisely two focus-focus singularities. Since the twisting index (one of the semitoric invariants) is related to the relationship between different focus-focus points, this paper provides systems for the future study of the twisting index.where ω S 2 is the standard volume form on the sphere and 0 < R 1 < R 2 are real numbers. For R := (R 1 , R 2 ) and t := (t 1 , t 2 , t 3 , t 4 ) ∈ R 4 define J R , H t : M → R by (1) where (x i , y i , z i ) are Cartesian coordinates on S 2 ⊂ R 3 for i = 1, 2. Then there exist choices of t 1 , t 2 , t 3 , t 4 , R 1 , R 2 such that (M, ω, (J R , H t )) is a semitoric system with exactly two focus-focus points.Theorem 1.1 is restated in more detail in Section 3 as Theorem 3.1. The coupled angular momenta system with coupling parameter t ∈ ]0, 1[ is the special case of Equation (1) with t 1 = t, t 3 = t 4 = 1−t, and t 2 = 0. The coupled angular momenta system describes the rotation of two vectors (with magnitudes R 1 and R 2 ) about the z-axis and has as a second integral a linear combination of the z-component of the first vector and the inner produce of the two vectors, while the system in Equation (1) includes additionally the z-component of the second vector and also

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Inverse spectral theory for semiclassical Jaynes–Cummings systems

Cited by 22 publications

References 40 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

Integrable systems, symmetries, and quantization

A family of compact semitoric systems with two focus-focus singularities

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