The quantum Rabi model: towards Braak’s conjecture

We establish a density one version of Braak’s conjecture on the fine structure of the spectrum of the quantum Rabi model, as well as a recent conjecture of Braak, Nguyen, Reyes-Bustos and Wakayama on the nearest neighbor spacings of the spectrum. The proof uses a three-term asymptotic expansion for large eigenvalues due to Boutet de Monvel and Zielinski, and a number theoretic argument from uniform distribution theory.


Introduction
In this note we address a conjecture of Braak [1] about the fine structure of the spectrum of the quantum Rabi model (QRM), a fundamental model of light-matter interaction, which describes the interaction between a two-level atom (qubit) coupled to a quantized, single-mode harmonic oscillator, see the survey [7].
The Hamiltonian of the system is where σ x = ( 0 1 1 0 ), σ z = ( 1 0 0 −1 ) are the Pauli matrices of the two-level system, assumed to have level splitting 2∆; a † and a are the creation and annihilation operators of the harmonic oscillator with frequency set to be unity; and g > 0 measures the strength of the coupling between the systems.
The Rabi Hamiltonian commutes with a parity operator P = (−1) a † a σ z , and hence the Hilbert space of states decomposes into the ±1-eigenspaces of P which are preserved by H.The Rabi model Hamiltonian in each of the parity eigenspaces can be described by the Jacobi matrices The spectrum of H breaks up into a union of two parity classes.The spectrum in each parity class is non-degenerate, and this allows a unique labeling of the corresponding eigenvalues in increasing order . .and likewise for the negative parity class {E − n }.The eigenvalues in each parity class satisfy E ± n = n − g 2 + o(1) as n → ∞ [6,8], so that for n sufficiently large, each interval [n, n + 1] contains at most 4 shifted eigenvalues E ± n + g 2 .Braak [1] conjectured that Conjecture 1.1 (Braak's G-conjecture).For a given parity class, all intervals [n, n + 1] contains at most two shifted eigenvalues, two intervals containing no shifted eigenvalues are not adjacent, and two intervals containing two shifted eigenvalues are also not adjacent.
In this note, we show that Braak's conjecture holds for "almost all" n, in the following sense: Theorem 1.2.Fix ∆ > 0 and g > 0. For all but at most O(N 3/4+o (1) ) values of n ≤ N, the interval (n, n + 1) contains exactly two shifted eigenvalues of one of the parity classes, and none for the other parity class, while the adjacent intervals (n − 1, n) and (n + 1, n + 2) contain exactly two eigenvalues of the other parity class and none of the first parity class.Moreover, neither n nor n ± 1 are shifted eigenvalues.
In particular, almost all intervals [n, n + 1] contain exactly two elements of the shifted spectrum.
Concerning the last assertion, there are special choices of the parameters g and ∆ for which there are "exceptional" eigenvalues E such that E + g 2 is an integer, see [7, §3.2] and the references therein, and our theorem excludes n − g 2 being one of these eigenvalues for almost all n.
An application of Theorem 1.2 is to prove a recent conjecture of Braak, Nguyen, Reyes-Bustos and Wakayama [2] on the nearest neighbor spacings of the full spectrum.Denote by {E k } the ordered eigenvalues of H of both parity classes: In [2], the nearest neighbor spacings δ n := E n+1 − E n were classified into three types: positive if both E n , E n+1 fell into the positive parity class, negative if both fell into the negative parity class, and mixed if one of the pair was positive and one negative.Based on numerical observation, it was conjectured [2, eq 14] that Conjecture 1.3 (Spacings conjecture for the QRM).The frequencies of the three different types of nearest neighbor spacings are 1/4,1/4,1/2, respectively.
This clearly follows from the full conjecture of Braak, but since we establish that Braak's conjecture holds for 100% of n ′ s, we have also established Conjecture 1.3.
Finally, we examine the value distribution of the normalized deviations As an application of the method of proof of Theorem 1.2, we show that the deviations in each parity class satisfy an arcsine law: .
The proof of Theorem 1.2 starts with an approximation to the eigenvalues due to Boutet de Monvel and Zielinski [3] and concludes with a number-theoretic argument.
Acknowledgement: I thank Daniel Braak, Eduard Ianovich and Masato Wakayama for helpful discussions, and a referee for corrections to an earlier version of the paper.
This research was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 786758).

The case of good n's
Boutet de Monvel and Zielinski [3] proved a three term expansion for the eigenvalues in each parity class: (1) where (this approximation was apparently proposed in [4], see also [8]).

Bounding the exceptional set
To conclude the proof of Theorem 1.2, we need to bound the number of "bad" n ∈ An elementary argument due to Fejér (1920) (see e.g.[5, Chapter 1 §2]) shows that for any a > 0, and any shift γ ∈ R, for suitable c = c(a, γ) > 0, for N ≫ 1, for any interval and in particular, for an interval of length > N −1/2+o (1) the number of fractional parts which fall into that interval is asymptotically N/2 times the length of that interval.In our case, the length of the interval .
The argument for δ + n is identical.