Optimal gauge for the multimode Rabi model in circuit QED

In circuit QED, a Rabi model can be derived by truncating the Hilbert space of an anharmonic qubit coupled to a resonator. This truncation breaks the gauge invariance present in the full Hamiltonian. Here we derive a simple criterion for an optimal gauge such that the differences between the truncated and the full Hamiltonian are minimized. We find that it is determined by the ratio of the anharmonicity of the qubit to an averaged resonator frequency. We demonstrate that the usual choices of flux and charge gauge are not necessarily the preferred options in the case of multiple resonator modes.