Universal spectral bounds for the quantum Rabi model: Extending Braak's conjecture

The quantum Rabi model is a paradigmatic example of a minimal yet nontrivial light-matter interaction, whose spectrum is transcendental yet exhibits a number of regularities. Braak observed that the eigenvalues bunch or anti-bunch following strict rules, leading to a conjecture that links integrability in quantum systems and residual order in their spectra. While a general proof remains elusive, understanding this structure is crucial for distinguishing deterministic quantum dynamics from chaotic behavior. Here, we extend Braak’s conjecture through a set of eigenvalue inequalities. We prove the extended conjecture across low and intermediate splitting regimes, and provide universal upper bounds on the entire spectrum. Our results uncover additional layers of spectral organization in the quantum Rabi model and expand the analytic toolkit for strongly coupled quantum systems.

💡 Research Summary

The paper addresses the long‑standing problem of characterizing the spectrum of the quantum Rabi model (QRM), a minimal yet non‑trivial light‑matter interaction described by the Hamiltonian
(H_R = \omega a^\dagger a + g \sigma_x (a + a^\dagger) + \Delta \sigma_z).
Because the model is integrable only in a subtle sense, the eigenvalues are transcendental functions of the parameters and no closed‑form expressions exist. Braak’s 2011 conjecture identified a remarkable regularity: in each parity sector the eigenvalues (E_n^\pm) lie in the intervals (