bibkey: reyesbustos2026twophotonrabi
authors: Cid Reyes-Bustos, Masato Wakayama
year: 2026
title: "Two-photon quantum Rabi models – Spectral degeneracy and symmetries"
doi: 10.48550/arXiv.2609.00750
url: https://arxiv.org/abs/2609.00750v1
claim: "The paper studies the spectrum of the two-photon asymmetric quantum Rabi model, proves divisibility relations between constraint polynomials for shared-parity degeneracies, and conjectures that the constraint polynomial at the critical coupling x = 1 factors as a product of linear factors y + 2n(2n + 2rho - 1) and that for x > 1 it has positive coefficients."
strata_touched:
  - D5/S3/Quantum/Dynamics/TwoPhotonRabiConstraintPolynomials
license: citation-only
triage: anchor

Two-photon quantum Rabi models – Spectral degeneracy and symmetries

Reyes-Bustos and Wakayama study the two-photon asymmetric quantum Rabi model with coupling g, level splitting 2Δ and bias ε ∈ ℝ, with parity ρ ∈ {0, 1}. Juddian (polynomial) eigenfunctions exist exactly when a constraint polynomial vanishes. Definition 4.1 gives these polynomials in x = (2g)^2 and y = Δ^2 by the three-term recursion

P_0 = 1, P_1 = y + 2x(4N + 2ρ + 2ε − 1) − 4(1 + ε), P_k = (y + 2xk(4N + 2ρ − 2k + 2ε + 1) − 4k(k + ε)) P_{k−1} − 4k(k − 1)(2(N − k + 1) + ρ)(2(N − k + 1) + ρ − 1) x P_{k−2},

and calls P_N^{(N,ρ,±ε)} the constraint polynomial. Proposition 4.4 records that P_N^{(N,ρ,±ε)}(1, y) does not depend on ε. After leaving “a complete description of this as an open problem”, the paper states Conjecture 4.5:

We have P_N^{(N,ρ,ε)}(1, y) = ∏_{n=1}^{N} (y + 2n(2n + 2ρ − 1)). Moreover, for x > 1, the polynomial P_N^{(N,ρ,ε)}(x, y) has positive coefficients, and therefore, no positive roots for y.

The critical value x = 1 corresponds to g = 1/2, where the model acquires continuous spectrum. In the paper’s own examples at x = 1, the product printed for P_5^{(5,1,ε)} carries the roots of the case ρ = 0.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2609.00750
• URL: https://arxiv.org/abs/2609.00750v1
• Version and location: arXiv:2609.00750v1 (2026-09-01, the only version), source file main.tex: §2 for the bias ε ∈ ℝ; §4 for Definition 4.1 (dfn:copo), Proposition 4.4 and Conjecture 4.5 (the conject environment numbered with the theorem counter within §4).