Addendum to ‘Algebraic equations for the exceptional eigenspectrum of the generalized Rabi model’

Abstract

In our recent paper (Li and Batchelor 2015 J. Phys. A: Math. Theor. 48 454005) we obtained exceptional points in the eigenspectrum of the generalized Rabi model in terms of a set of algebraic equations. We also gave a proof for the number of roots of the constraint polynomials defining these exceptional solutions as a function of the system parameters and discussed the number of crossing points in the eigenspectrum. This approach however, only covered a subset of all exceptional points in the eigenspectrum. In this addendum, we clarify the distinction between the exceptional parts of the eigenspectrum for this model and discuss the subset of exceptional points not determined in our paper.

A generalization of the quantum Rabi model: exact solution and spectral structure

“…For a discussion of the exceptional spectrum in the case of the original quantum Rabi model, see [35,27]; for a discussion of the exceptional spectrum of a different generalization of the quantum Rabi model, obtained by adding an asymmetric term σ x , see [36,37]. A detailed mathematical symmetry analysis using Lie algebra representations of sl 2 (R) is given for the spectrum of the original quantum Rabi model in [38] and of the asymmetric quantum Rabi model in [39].…”

Generalized adiabatic approximation to the quantum Rabi model

“…As pointed out in the Introduction, there are no simple closed-form solutions to the regular spectrum of the QRM. Instead, only the level crossing points in the spectrum can be determined through finite-term constraint polynomials of system parameters [30,31]. For this reason, the QRM is also referred to as quasi-solvable [32][33][34].…”

The quantum Rabi model: solution and dynamics

“…The other, non-degenerate, set of exceptional points have also been discussed [40,51,86,89,90]. The system parameters do not satisfy a constraint polynomial at these points, but other conditions can be derived.…”

“…The class of exceptional points not satisfying a constraint polynomial have also been discussed for this model [90]. It has been shown that the effect of the bias parameter ǫ is to induce a conical intersection point in the energy spectrum at each of the two-fold degenerate Judd points located at ǫ = 0 [138].…”