## 2 Structures and Models  
The model system for RBO observation in the strong coupling regime is shown in Figure 1. It corresponds to the periodic two-level atomic chain, excited with an obliquely incident ac field in the strong coupling regime, and driven with dc voltage applied at the ends. The neighboring atoms are coupled via interatomic tunneling, with different values of penetration at the ground and excited states. The model system for RBO analysis in the ultrastrong coupling regime is shown in Figure 2. Analysis of RO in the regime of ultrastrong coupling was given in [43], performing on a model system depicted in Figure 2a, where a mode of microcavity of THz frequency range is coupled with the electronic excitation of a highly doped quantum well (QW) inserted inside the capacitor. There are at least two confined subbands in the QW, and the fundamental subband hosts a two-dimensional electron gas with an arbitrary total number of electrons. The energy difference between the first and the second subband is matched with the resonance of the LC circuit. The reduction of the capacitor armatures also implies a reduction of the capacitance C, which in turn changes the resonant frequency of the LC circuit. In order to keep the resonator frequency always matched with the frequency of the quantum transition, one has to increase the inductance, L. The strong reduction of the capacitor volume in the quantum LC resonator allows for the realization of the ultrastrong coupling regime with a few electrons only, even with a single electron. It was focused in [43] on the case of an electron gas coupled with the circuit photons of the capacitor. One of the peculiar features of such a quantum mechanical system is the existence of entangled light-matter excitations in analogy with the super-radiant and sub-radiant Dicke states in atomic clouds [43]. Figure 1: (**a**) General illustration of the periodic two-level atomic chain used as a model, indicating Rabi–Bloch oscillations (RBO) in strong coupling regime. It is excited with obliquely incident ac field, and driven with dc voltage applied at the ends. The neighboring atoms are coupled via interatomic tunneling, with different values of penetration at the ground and excited states; (**b**) Ground and excited energy levels of single two-level atom separated by the transition energy \(\hbar\omega_{0}\). Figure 2: General illustration of the periodic chain used as a model, indicating RBO in ultrastrong coupling regime. (**a**) The single element of the chain. It corresponds to a highly doped quantum well (QW) inserted inside the capacitor of microcavity of THz frequency range [43]. It is resonantly coupled with a mode of microcavity. The frequency matching is reached via the choice of inductance value; (**b**) A highly doped QW inserted inside the capacitors of the chain of coupled microcavities. The dc voltage applied to the system at the ends of the chain. The THz oscillating voltage is applied through a nano-antenna; (**c**) The periodic potential induced by additional doping for transforming the homogeneous QW to the effective superlattice. The coupling is governed by the inter-barrier electron tunneling. The system under consideration in our case, is in essence, different. We consider the periodic chain of elements [43], every one of which plays the role of the single atom in Figure 1. The chain is driven by both dc and classical THz frequency field (Figure 2b). The THz oscillating voltage is applied to the system through a nano-antenna [54; 55; 56; 57; 58; 59; 60; 61; 62; 63] (Figure 2b). The QW is assumed to be periodically doped, thus the additional periodic potential has been induced (Figure 2c). We assume only one electron in the system. The coupling is governed by the inter-barrier electron tunneling. In our case, QW represents the effective super-lattice, which is able to support the electron's BO [21] and was the first system in which BO was experimentally observed [21; 41]. Of course, these two models are identical from a physical point of view.