Optimal Dynamical Gauge in the Quantum Rabi Model

We investigate the gauge dependence of physical observables in the quantum Rabi model (QRM) under di erent potential fields, arising from the Hilbert-space truncation of the atomic degrees of freedom. In both square-well and harmonic potentials, we find that the optimal gauge for the groundstate energy depends on the cavity frequency: the dipole gauge is optimal in the low-frequency limit, while the Coulomb gauge is optimal in the high-frequency limit. For a dynamical quantity such as the out-of-time-order correlator (OTOC), we demonstrate the necessity of introducing an optimal dynamical gauge. This study provides deeper insights into the intricate relationship between gauge choice and the dynamics of quantum electrodynamics (QED) systems, paving the way for more accurate theoretical frameworks.