  Strong coupling between a single resonator mode and a single quantum emitter
is key to a plethora of experiments and applications in quantum science and
technology and is commonly described by means of the Jaynes-Cummings model.
Here, we show that the Jaynes-Cummings model only applies when the cavity does
not significantly change the emitter's emission rate into free-space. Most
notably, the predictions made by the Jaynes-Cummings model become increasingly
wrong when approaching the ideal emitter-resonator systems with no free-space
decay channels. We present a Hamiltonian that provides, within the validity
range of the rotating wave approximation, a correct theoretical description
that applies to all regimes. As minimizing the coupling to free-space modes is
paramount for many cavity-based applications, a correct description of strong
light-matter interaction is therefore crucial for developing and optimizing
quantum protocols.
