Bistability effect in the extreme strong coupling regime of the Jaynes-Cummings model

We study the nonlinear response of a driven cavity QED system in the extreme strong coupling regime where the saturation photon number is below one by many orders of magnitude. In this regime, multi-photon resonances within the Jaynes--Cummings spectrum up to high order can be resolved. We identify an intensity and frequency range of the external coherent drive for which the system exhibits bistability instead of resonant multi-photon transitions. The cavity field evolves into a mixture of the vacuum and another quasi-classical state well separated in phase space. The corresponding time evolution of the outgoing intensity is a telegraph signal alternating between two attractors.