Out-of-time-order correlator in the quantum Rabi model

We investigate signatures of chaos and equilibration in the quantum Rabi model, which exhibits a quantum phase transition when the ratio of the atomic level-splitting to bosonic frequency grows to infinity. We show that the out-of-time-order correlator derived from the Loschmidt echo signal quickly saturates in the normal phase and undergoes exponential growth in the superradiant phase, which is associated with the onset of quantum chaos. Furthermore, we show that the effective time-averaged dimension of the quantum Rabi system can be large compared with the spin system size, which leads to suppression of the temporal fluctuations and equilibration of the spin system.