When the vacuum Rabi frequency, Ω , of an electromagnetic mode is much smaller than the bare frequency of the excitation to whom it couples, ω, the simple Jaynes-Cumming or Tavis-Cumming models capture the main features of light-matter interaction and cavity QED1,2. However, already for a normalised coupling > . ω Ω 0 1, the Rotating Wave Approximation (RWA) that justifies those solvable models...