# Symmetries in the Quantum Rabi Model

## Abstract

**:**

## 1. Introduction

## 2. The Rotating-Wave Approximation and Its Symmetry

## 3. Integrability of Systems with Less Than Two Continuous Degrees of Freedom

## 4. The Global Spectrum of the QRM

**Conjecture**

**1.**

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) The QRM spectrum for $\omega =1$, $\Delta =0.7$ as function of the coupling constant g. Instead of the energy, the spectral parameter $x=E+{g}^{2}$ is displayed on the ordinate. States with negative (positive) parity are displayed in blue (red). Within the same parity subspace all level crossings are avoided (green circles). (

**Right**) The Jaynes–Cummings-spectrum for the same parameters. In this case, corresponding states do cross due to the enhanced symmetry of the JCM (small green circles).

**Figure 2.**The JCM spectrum at resonance $2\Delta =\omega =1$ as a function of g. Each color corresponds to an invariant subspace ${\mathcal{H}}_{n}$. The state $|\mathrm{vac}\rangle =|0\rangle \otimes |\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\downarrow \rangle $ spans the (trivial) irreducible representation of $U\left(1\right)$ with character 0.

**Figure 3.**(

**Left**) The QRM spectrum (blue) and the approximation by the GRWA (green) as function of g for $\Delta =0.7$. The GRWA reproduces the qualitative properties of the spectral graph also for large coupling. (

**Right**) The QRM and GRWA spectra as function of $\Delta $ for $g=0.25$. The blue (red) level lines correspond to negative (positive) parity in the QRM. In this case, the GRWA shows level crossings (small black circles) where the QRM has none (black circles) because there are no degeneracies for fixed parity. All apparent degeneracies of the QRM within the same parity chain are narrow avoided crossings.

**Figure 4.**The distribution of level distances $\Delta E={E}_{n+1}-{E}_{n}$ of the QRM for positive parity as function of the level number n. Parameters are $\omega =1$, $g=\Delta =5$. A clear deviation from the exponential law predicted in [32] is visible.

**Figure 5.**(

**Left**) The energy shell $[E,E+\delta E]$ (red lines) contains the integer-valued vectors $({n}_{1},{n}_{2})$ (blue crosses) belonging to the quantization of the action variables ${I}_{1}=\hslash ({n}_{1}+{\alpha}_{1}/4)$ and ${I}_{2}=\hslash ({n}_{2}+{\alpha}_{2}/4)$. The distance of adjacent energies $\tilde{f}({n}_{1},{n}_{2})-\tilde{f}({n}_{1}^{\prime},{n}_{2}^{\prime})$ is statistically unrelated for large quantum numbers if $\tilde{f}$ is non-linear. (

**Right**) If the second action variable ${I}_{2}$ can take only two values as would be the case for a discrete degree of freedom with dim$\mathcal{H}=2$, the average level distance is the same as for linear $\tilde{f}$ and Poisson statistics does not apply.

**Figure 6.**(

**Left**) The Dicke spectrum for spin 3/2 and $\Delta =0.7$ as function of g. The spectral parameter is $x=E+{g}^{2}/3$, making the “baselines of the first kind” [27] horizontal while the baselines of the second kind are given as dashed lines. (

**Right**) The QRM spectrum at $\Delta =0.7$ for comparison ($x=E+{g}^{2}$). All level crossings are located on the horizontal baselines with $x=\mathrm{const}$.

**Figure 7.**(

**Left**) ${G}_{+}\left(x\right)$ and its entire approximation ${G}_{+}(x;0.4,0)$ for $\Delta =1$. (

**Right**) ${G}_{+}\left(x\right)$ for $\Delta =1$ (blue), 2 (red), 4 (green) and 7 (orange).

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Braak, D.
Symmetries in the Quantum Rabi Model. *Symmetry* **2019**, *11*, 1259.
https://doi.org/10.3390/sym11101259

**AMA Style**

Braak D.
Symmetries in the Quantum Rabi Model. *Symmetry*. 2019; 11(10):1259.
https://doi.org/10.3390/sym11101259

**Chicago/Turabian Style**

Braak, Daniel.
2019. "Symmetries in the Quantum Rabi Model" *Symmetry* 11, no. 10: 1259.
https://doi.org/10.3390/sym11101259