Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model

Li, Zi-Min; Batchelor, Murray T.

doi:10.1088/1751-8113/48/45/454005

Cited by 52 publications

(137 citation statements)

References 30 publications

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“…[10], proved at different levels of generality in Refs. [19,20] and supported by the results of numerical studies [14,20]. This situation raises the question of what symmetry might exist in the system and hence explain the observed energy level crossings.…”

Section: Introductionmentioning

confidence: 79%

Attempt to find the hidden symmetry in the asymmetric quantum Rabi model

Ashhab

2020

Phys. Rev. A

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It has been observed that the asymmetric quantum Rabi model (QRM), which does not possess any obvious symmetry, exhibits energy levels crossings, which are often associated with symmetries. This observation suggests that there is in fact a symmetry in the asymmetric QRM, even though simple inspection of the model and its Hamiltonian does not reveal the nature of this symmetry. Here we present the results of numerical calculations on the energy eigenstates of the asymmetric QRM in an attempt to elucidate the nature of the symmetry. In particular, we note that the distribution of states in the Hilbert space among different symmetry classes is normally independent of system parameters and test whether this property holds for the asymmetric QRM. We find that it does not, which both helps explain why the symmetry is hidden and adds more intrigue as to what its nature might be.

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Section: Introductionmentioning

confidence: 79%

Attempt to find the hidden symmetry in the asymmetric quantum Rabi model

Ashhab

2020

Phys. Rev. A

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“…The constraint polynomials P n (x, y) for the AQRM were defined in [12] following the work of Kús [25] on the (symmetric) quantum Rabi model. These polynomials were derived in the framework of finite-dimensional irreducible representations of sl 2 in the confluent Heun picture of the AQRM [13].…”

Section: Constraint Polynomialsmentioning

confidence: 99%

“…The full eigenspectrum can be determined from the analytical solution. The exceptional parts, known as Juddian isolated exact solutions [11], can be systematically found from the conditions under which the confluent Heun functions are terminated as finite polynomials [6,12]. The eigenvalues are simply those of a shifted oscillator, however with the system parameters satisfying constraint polynomials which become increasingly complicated for higher energy levels.…”

Section: Introductionmentioning

confidence: 99%

The asymmetric quantum Rabi model and generalised Pöschl–Teller potentials

Guan¹,

Li²,

Dunning³

et al. 2018

J. Phys. A: Math. Theor.

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Starting with the Gaudin-like Bethe ansatz equations associated with the quasi-exactly solved (QES) exceptional points of the asymmetric quantum Rabi model (AQRM) a spectral equivalence is established with QES hyperbolic Schrödinger potentials on the line. This leads to particular QES Pöschl-Teller potentials. The complete spectral equivalence is then established between the AQRM and generalised Pöschl-Teller potentials. This result extends a previous mapping between the symmetric quantum Rabi model and a QES Pöschl-Teller potential. The complete spectral equivalence between the two systems suggests that the physics of the generalised Pöschl-Teller potentials may also be explored in experimental realisations of the quantum Rabi model.

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“…In particular, sl 2 algebra does not explain why the Juddian isolated exact solutions can be analytically computed [9,10,[13][14][15], whereas the remaining part of the spectrum not. The…”

Section: Introductionmentioning

confidence: 99%

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

Moroz¹

2018

J. Phys. A: Math. Theor.

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General concept of a gradation slicing is used to analyze polynomial solutions of ordinary differential equations (ODE) with polynomial coefficients, Lψ = 0, where L = l p l (z)d l z , p l (z) are polynomials, z is a one-dimensional coordinate, and d z = d/dz. It is not required that ODE is ei-Fuchsian or (ii) leads to a usual Sturm-Liouville eigenvalue problem. General necessary and sufficient conditions for the existence of a polynomial solution are formulated involving constraint relations. The necessary condition for a polynomial solution of nth degree to exist forces energy to a nth baseline. Once the constraint relations on the nth baseline can be solved, a polynomial solution is in principle possible even in the absence of any underlying algebraic structure. The usefulness of theory is demonstrated on the examples of various Rabi models. For those models, a baseline is known as a Juddian baseline (e.g. in the case of the Rabi model the curve described by the nth energy level of a displaced harmonic oscillator with varying coupling g). The corresponding constraint relations are shown to (i) reproduce known constraint polynomials for the usual anddriven Rabi models and (ii) generate hitherto unknown constraint polynomials for the two-mode, two-photon, and generalized Rabi models, implying that the eigenvalues of corresponding polynomial eigenfunctions can be determined algebraically. Interestingly, the ODE of the above Rabi models are shown to be characterized, at least for some parameter range, by the same unique set of grading parameters.

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Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model

Cited by 52 publications

References 30 publications

Attempt to find the hidden symmetry in the asymmetric quantum Rabi model

Attempt to find the hidden symmetry in the asymmetric quantum Rabi model

The asymmetric quantum Rabi model and generalised Pöschl–Teller potentials

A unified treatment of polynomial solutions and constraint polynomials of the Rabi models

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