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The density conjecture for Juddian points for the quantum Rabi model

Rishi Kumar, Zeév Rudnick

TL;DR

This work advances the mathematical understanding of the quantum Rabi model by proving a strong form of the density conjecture for Juddian (degenerate) eigenvalues when the level splitting $2\u0394$ is fixed. The authors connect the constraint polynomials $P_N(X,Y)$ to classical Laguerre polynomials via $P_N(x,0)=(-1)^N(N!)^2 L_N(x)$ and exploit a determinantal tri-diagonal representation to relate zeros of $P_N$ to eigenvalues of a symmetric matrix, enabling Weyl perturbation and interlacing analyses. They establish a precise asymptotic density: for fixed $\u0394>0$, the number of Juddian eigenvalues with $g\le\Gamma$ among $N$-indexed constraints scales as $\sim (4/\pi)\Gamma\sqrt{N}$, and show dense occurrence of Juddian points in the coupling parameter. Moreover, they prove the existence of infinitely many parameter choices yielding two distinct Juddian eigenvalues by analyzing intersections of zero loci $Z_m$ and $Z_N$, leveraging the distribution of Laguerre zeros (via Gawronski) and the branch structure of $Z_n$. The results underscore the central role of Laguerre zeros in the fine structure of spectral degeneracies for the QRM and open pathways to further understanding multi-Juddian phenomena.

Abstract

We study doubly degenerate (Juddian) eigenvalues for the Quantum Rabi Hamiltonian, a simple model of the interaction between a two-level atom and a single quantized mode of light. We prove a strong form of the density conjecture of Kimoto, Reyes-Bustos, and Wakayama, showing that any fixed value of the splitting between the two atomic levels, there is a dense set of coupling strengths for which the corresponding Rabi Hamiltonian admits Juddian eigenvalues. We also construct infinitely many sets of parameters for which the Rabi Hamiltonian admits two distinct Juddian eigenvalues. The fine structure of the zeros of classical Laguerre polynomials plays a key role in our methods.

The density conjecture for Juddian points for the quantum Rabi model

TL;DR

This work advances the mathematical understanding of the quantum Rabi model by proving a strong form of the density conjecture for Juddian (degenerate) eigenvalues when the level splitting is fixed. The authors connect the constraint polynomials to classical Laguerre polynomials via and exploit a determinantal tri-diagonal representation to relate zeros of to eigenvalues of a symmetric matrix, enabling Weyl perturbation and interlacing analyses. They establish a precise asymptotic density: for fixed , the number of Juddian eigenvalues with among -indexed constraints scales as , and show dense occurrence of Juddian points in the coupling parameter. Moreover, they prove the existence of infinitely many parameter choices yielding two distinct Juddian eigenvalues by analyzing intersections of zero loci and , leveraging the distribution of Laguerre zeros (via Gawronski) and the branch structure of . The results underscore the central role of Laguerre zeros in the fine structure of spectral degeneracies for the QRM and open pathways to further understanding multi-Juddian phenomena.

Abstract

We study doubly degenerate (Juddian) eigenvalues for the Quantum Rabi Hamiltonian, a simple model of the interaction between a two-level atom and a single quantized mode of light. We prove a strong form of the density conjecture of Kimoto, Reyes-Bustos, and Wakayama, showing that any fixed value of the splitting between the two atomic levels, there is a dense set of coupling strengths for which the corresponding Rabi Hamiltonian admits Juddian eigenvalues. We also construct infinitely many sets of parameters for which the Rabi Hamiltonian admits two distinct Juddian eigenvalues. The fine structure of the zeros of classical Laguerre polynomials plays a key role in our methods.
Paper Structure (10 sections, 13 theorems, 63 equations, 3 figures)

This paper contains 10 sections, 13 theorems, 63 equations, 3 figures.

Table of Contents

  1. Introduction
  2. About the proofs
  3. PRELIMINARY RESULTS
  4. Background on constraint polynomials
  5. A matrix representation
  6. Proof of the density conjecture
  7. Proof of Theorem \ref{['thm:densityJudd']}
  8. Proof of Theorem \ref{['thm:Juddv1m']}
  9. Properties of $Z_m$
  10. Proof of Theorem \ref{['thm:Juddv1m']}

Key Result

Theorem 1.1

Fix the level splitting $2\Delta>0$. Then, the set of coupling constants $g>0$ for which $H_{g,\Delta}$ admits Juddian eigenvalues is dense in the set of all possible coupling constants. Moreover, we have a limit density: For any fixed $\Gamma>0$, as $N\to \infty$,

Figures (3)

  • Figure 1: The functions $G^+(z)$ (red) and $G^-(z)$ (blue, dashed) for $g=0.7$ and $\Delta=0.4$ (\ref{['figsub:GBraak']}), and for $g=1/ \sqrt{8}$, $\Delta=1/\sqrt{2}$ (\ref{['figsub:GfunctionsJudd']}), when the pole at $x=1$ is cancelled and $E= 1-g^2$ is an exceptional (Juddian) eigenvalue.
  • Figure 2: The zero locus $P_3(x,y)=0$. There are three branches, the $m$-th connecting the point $(m^2,0)$ to $(0,\lambda_{3,m})$ where $0<\lambda_{3,1}<\lambda_{3,2} <\lambda_{3,3}$ are the zeros of the Laguerre polynomial $L_3(x)$. The line $y=0.5$ intersects the zero locus in three points, all in the positive quadrant.
  • Figure 3: The zero set $Z_{20}$ of the constraint polynomial $P_{20}(x,y)$ (left). The intercepts with the $y$-axis are at $y=m^2$, $m=1,\dots,20$, and the intercepts with the $x$-axis are at the zeros of the Laguerre polynomial $L_{20}(x)$, which are $\lambda_{20,1}=0.0705399$, $\lambda_{20,2}=0.372127$, $\lambda_{20,3}=0.916582$, $\ldots, \lambda_{20,20} = 66.5244$. On the right, the region $0<x<1$, $0<y<2$, with $Z_1=\{x+y-1=0\}$ (dashed, red).

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 9 more