Integrability of the Rabi Model

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.

Figure 1

$G_{+}(x)$ [light gray (red) lines] and $G_{-}(x)$ [dark gray (blue) lines] in the interval [$-1,5$] for $\omega =1$, $g=0.7$, and $\Delta =0.4$

Figure 2

Rabi spectrum for $\Delta =0.4$, $\omega =1$, and $0\le g\le 0.8$ in the spaces with positive [light gray (red) lines] and negative [dark gray (blue) lines] parity. Within each space the states are labeled with ascending numbers $0,1,2,\dots$. This labeling does not change with $g$ because no lines intersect within spaces of fixed parity. The spectral graph is composed of two intersecting “ladders” of level lines, each corresponding to one parity subspace. This labeling is used on the right side of the figure. On the left side the states with $g=0$ are labeled by the uncoupled d.o.f.; i.e., in $|\pm ,n⟩$, $+$ or $-$ corresponds to the two-level system and $n=0,1,2,\dots$ to the eigenstates of the bosonic mode.

Figure 3

The spectrum of the Jaynes-Cummings model (black lines) compared with the Rabi model [gray (red) lines] for $\Delta =0.4$ and even parity. The state labeling of the former has the form $|\psi ⟩=|n_0,C⟩$ with $n_0=\pm$. Two accidental crossings occur between levels with $C=5$ and $C=3$ at $g\approx 0.5$ and between $C=1$ and $C=3$ at $g\approx 0.73$.

Figure 4

The spectrum of the generalized Rabi model with broken ${\mathbb{Z}}_2$ symmetry and $\Delta =0.7$, $ϵ=0.2$. The twofold labeling of the states corresponds to the two integrable limits $g=0$ on the left and $g\to \infty$ on the right. For finite $g$ neither labeling classifies the states properly.