Renormalization Group Flow of the Jaynes-Cummings Model

The Jaynes-Cummings model is a cornerstone of light-matter interactions. While finite, the model provides an illustrative example of renormalization in perturbation theory. We show, however, that exact renormalization reveals a rich nonperturbative structure, and that the model provides a physical example of a theory with a chaotic coupling trajectory and multivalued $\beta$ function. We also construct an exact Wilsonian-like renormalization group flow for the effective scattering matrix, and show how multivalued features arise in the flow. Our results shed light on nonperturbative aspects of renormalization and on the structure of the Jaynes-Cummings model itself.

Figure 1

Scattering probabilities ${\mathbb{P}}_j$ (columns) given by different branches of the solution to the renormalization condition at ${\mathbb{P}}_{\mathrm{obs}}=1/10$. All branches [rows, with $n$ and $\pm$ labeling the choices in (6)] yield the same ${\mathbb{P}}_0={\mathbb{P}}_{\mathrm{obs}}=1/10$ by construction, but give different ${\mathbb{P}}_j$ for $j>0$. The fact that the rows differ shows that different branches define different theories.

Figure 2

Starting from $g_r\gtrsim 0$, the coupling flows toward $g_r=1$, where the $\beta$ function (9) switches branch, and the flow turns toward $g_r=-1$, and so on. The flow is shown after encountering $n$ turning points, for $n$ from 0 to 4.

Figure 3

Coupling trajectories as a function of RG time. Both exact solutions (solid lines) yield the same renormalized coupling $g_r(\mathsf{0})=1/2$, but come from different branches of (6) corresponding to $g_0=\pi /6+2\pi$ and $g_0=\pi /6+12\pi$. The 1-loop trajectories (dashed lines) tend to the fictitious fixed point at $g_r=\sqrt{3}$.

Figure 4

Solutions to the renormalization condition (13) defining the coupling $e_k$, at $g_r=1/2$. Points show numerical solutions. As $k$ decreases toward 1, multiple solutions appear in the flow, and interpolate toward the IR solutions (14) (horizontal lines) shown for $n=0,\dots ,4$.