Strong and ultra-strong light-matter coupling are remarkable phenomena of quantum electrodynamics occurring when the interaction between a matter excitation and the electromagnetic field cannot be described by usual perturbation theory. This is generally achieved by coupling an excitation with large oscillator strength to the confined electromagnetic mode of an optical microcavity. In this work we demonstrate that strong/ultra-strong coupling can also take place in the absence of optical confinement. We have studied the non-perturbative spontaneous emission of collective excitations in a dense two-dimensional electron gas that superradiantly decays into free space. By using a quantum model based on the input-output formalism, we have derived the linear optical properties of the coupled system and demonstrated that its eigenstates are mixed light-matter particles, like in any system displaying strong or ultra-strong light-matter interaction. Moreover, we have shown that in the ultra-strong coupling regime, i.e. when the radiative broadening is comparable to the matter excitation energy, the commonly used rotating-wave and Markov approximations yield unphysical results. Finally, the input-output formalism has allowed us to prove that Kirchhoffs law, describing thermal emission properties, applies to our system in all the light-matter coupling regimes considered in this work.
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