The interaction of a two-level system (TLS) with a single bosonic mode is one of the most fundamental processes in quantum optics. Microscopically, it is described by the quantum Rabi model (QRM). Here, we propose an implementation of this model based on single trapped cold atoms. The TLS is implemented using atomic Zeeman states, while the atoms vibrational states in the trap represent the bosonic mode. The coupling is mediated by a suitable fictitious magnetic field pattern. We show that all important system parameters, i.e., the emitter-field detuning and the coupling strength of the emitter to the mode, can be tuned over a wide range. Remarkably, assuming realistic experimental conditions, our approach allows one to explore the regimes of ultra-strong coupling, deep strong coupling, and dispersive deep strong coupling. The states of the bosonic mode and the TLS can be prepared and read out using standard cold-atom techniques. Moreover, we show that our scheme enables the implementation of important generalizations, namely, the driven QRM, the QRM with quadratic coupling as well as the case of many TLSs coupled to one mode (Dicke model). The proposed cold-atom based implementation will facilitate experimental studies of a series of phenomena predicted for the QRM in extreme, so far unexplored physical regimes.
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