Identifying unbound strong bunching and the breakdown of the Rotating Wave Approximation in the quantum Rabi model

We use a recently derived gauge-invariant formulation of the problem of a two-level system coupled to an optical cavity, to explore the transition between the weak, and the ultra-strong coupling regimes of light-matter interaction. We explore this transition using the intensity correlations $g^{(2)}(\tau)$ of the emitted light, and find strong, unbounded bunching of the emission from systems governed by the Rabi Hamiltonian. Surprisingly, this effect is observed not only in the ultra-strong coupling regime, but also for weakly coupled systems, where the Jaynes-Cummings Hamiltonian would predict the opposite, antibunched emission. This suggests that the higher-order correlations are a particularly sensitive probe of the divergence between the Jaynes-Cummings and Rabi Hamiltonians, and can serve as an indicator of the breakdown of the rotating wave approximation. Our findings indicate also that the boundary between the weakly, strongly, and ultra-strongly coupled dynamics, is much richer than currently accepted.