# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 1 Introduction  
The early quantum theory of electrical conductivity in crystal lattices by [PERSON], [PERSON] and [PERSON] [1; 2; 3; 4] led to the prediction that a homogeneous dc field induces an oscillatory rather than uniform motion of the electrons. These so-called Bloch oscillations (BO) have been observed in bulk N-doped GaAs at a lattice temperature of 300 K in high fields up to 300 kV/cm [5], as well as different types of artificial systems such as semiconductor superlattices [6], interacting atoms in optical lattices [7; 8], ultracold atoms [9; 10; 11; 12; 13; 14], light intensity oscillations in waveguide arrays [15; 16; 17; 18; 19; 20; 21], acoustic waves in layered and elastic structures [22], and atomic oscillations in Bose-Einstein condensates [23] among others. Several recent studies have investigated the dynamics of cold atoms in optical lattices subject to ac forcing; the theoretically predicted renormalization of the tunneling amplitudes has been verified experimentally. The recent observations include global motion of the atom cloud, such as giant \"super-Bloch oscillations\" [24]. As a result, BO transformed from the specific contra intuitive model to the general experimentally-supported physical concept of oscillatory motion of wave packets placed in a periodic potential when driven by a constant force [8; 25]. Rabi oscillations (RO) are periodical transitions of a two-level quantum system between its stationary states under the action of an AC driving field [26; 27]. The phenomenon was theoreticallypredicted by [PERSON] for the nuclear spins in radio-frequency magnetic field [28] and afterwards, discovered in various physical systems, such as electromagnetically driven individual atoms [29] including the case of Rydberg atomic states [30], semiconductor quantum dots (QDs) [31] and different types of solid-state qubits (superconducting charge qubits based on Josephson junctions [32; 33; 34], spin qubits [35], semiconductor charge qubits [36]). The textbook picture of the Rabi effect is given by the Jaynes-Cummings model [26; 27]. It implied that the concept of the dressed atom with light was correspondent to the quantum entanglement of electrons and photons. This model can be essentially modified by a set of additional features, such as the broken inversion symmetry [37], the propagation of RO over the chains of coupled atoms in the form of special waves (Rabi-waves), and depolarization due to the local fields [38; 39; 40; 41; 42; 43]. It opens the possibility of the generation of a variety of entangled quantum states, which could have a great impact on the search for universal and efficient quantum computation processes [44], the electrically tunable optical nano-antennas with highly directive emission [39; 40; 43], quantum sensing [45], and metrology [46]. Recent theoretical progress in this area is associated, in particular, with the novelty of inter-atomic coupling mechanisms [47; 48], which manifests themselves in surprising thermodynamic behavior of the specific waveguiding arrays, which recently was experimentally verified [49]. In this paper, we build a theoretical model of a chain of coupled two-level quantum elements exposed to quantum light and driven via bias voltage. We consider the case of strong coupling of light with charge carriers, which lead to the entanglement of electron-photon quantum states. This model describes BO of electrons dressed with light and their mutual influence with RO. Our model has a significant degree of generality: It relates to the systems of different physical origin and various frequency ranges. We consider its application to semiconductor heterostructures (THz frequencies), semiconductor quantum dots (visible frequencies), and Josephson junctions (microwaves). For brevity, we refer to every of these artificial two-level quantum objects as an \"atom\" regardless of its physical implementation. We develop the model taking into account conditions of real experiments.