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

TL;DR

The paper extends Braak's conjecture for the quantum Rabi Hamiltonian by proposing universal spectral inequalities that tightly bound eigenvalues across parity sectors. It shows that inequalities $-\,\omega \le E^+_n - E^-_n \le \omega$, $E^pm_{n+1} - 2\omega \le E^pm_n$, and $E^pm_n \le (n+\1/2)\omega - g^2/\omega$ hold (with strict variants for $g\neq 0$), and proves the extended conjecture in low- and intermediate-splitting regimes $|\Delta| \le \omega/2$ while providing a universal upper bound on the spectrum via displacement methods, Weyl-type perturbation arguments, and Gershgorin-disk estimates. A harmonic-oscillator approximation in the strongly coupled widely-split regime clarifies eigenvector localization and energy shifts, linking the spectrum to a Lamb-shifted oscillator picture. Collectively, these results reveal additional layers of spectral organization in the quantum Rabi model and furnish analytic tools applicable to nonperturbative, integrable, or near-integrable quantum systems.

Abstract

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.

Figures (3)

• Figure 1: Product states between a two-level system (on the top) and a quantum harmonic oscillator (on the right) are represented as red and blue dots, arranged according to their factors (height indicates the $\left|n\right\rangle$ factor, left for $\left|\downarrow\right\rangle$, and right for an $\left|\uparrow\right\rangle$ factor) and connected according to the transitions allowed by the Hamiltonian Lanuza2024. The coupling $g$ between both subsystems generates two independent sectors $\mathcal{H}_\pm$.
• Figure 2: Differences between eigenvalues $E^\pm_n$ of the QRM, obtained numerically through a $50\times50$ truncation of $H_\pm$ for variable coupling $g$ and several level splittings $\Delta$ (color-coded according to the scale shown at the lower right). Gray areas are conjectured to be off-bounds for the corresponding energy differences.
• Figure 3: a) Shifted eigenvalues $x_n^+$ (in red) and $x_n^-$ (in blue) of Hamiltonian $H_\pm$ when $\Delta=4g$. The cyan dashed line corresponds with $\epsilon={\left(4 g^4-\Delta ^2 \omega ^2\right)^2}/\left({64 g^6 \omega ^2}\right)$ whereas cyan solid lines are drawn at $\epsilon=0,1,2,$... . b) Eigenvector amplitudes $a_n$ (when $n$ is even) and $b_n$ (for $n$ odd) of $H_+$ for parameter values $\Delta=-4g=80\omega$. Dots are obtained numerically from the three lowest eigenstates of a $600\times600$ truncation of the matrix representation \ref{['eq:Hpm']}; cyan diamonds are the corresponding amplitudes from the analytic approximation \ref{['eq:eigenstate']}; and the 3D shapes are the eigenstates of a quantum harmonic oscillator in position space.