The Jaynes-Cummings model maps exactly onto superconducting circuits at charge degeneracy

At the charge degeneracy point ($N_g=1/2$) of a Cooper pair box inside a transmission line resonator, the full circuit Hamiltonian reduces exactly to the Jaynes-Cummings Hamiltonian of atomic cavity QED. The qubit splitting $\Omega =E_J$ is set by the Josephson energy, and the coupling $g=(\beta e/\hslash )\sqrt{\hslash {\omega }_r/cL}$ is determined entirely by geometry and circuit parameters.

This is not an approximation or analogy — it is a mathematical identity. The artificial atom is permanently fixed at the field antinode (unlike real atoms transiting a cavity), so $g$ is constant and the system stays in strong coupling indefinitely.

The mapping breaks down away from degeneracy: a longitudinal coupling term $\propto (a^{†}+a){\sigma }_z$ appears, and $g$ is reduced by $\sin \theta$.

Source: blais-2004-circuit-qed Related: circuit-qed, cooper-pair-box-charge-qubit, transmon