# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 2 Statement of the Problem and Calculation Technique  
### Physical System and Model  
Let us consider a one-dimensional (1D) structure of identical atoms placed over a line with period \(a\) (see for example Figure 1a). Some other types of artificial atom chains are shown in Figures 2 and 3. Each atom is considered as a two-level Fermion system with transition frequency \(\omega_{0}\). The location of the atoms in the lattice points is determined by the radius-vector \(\mathbf{R}_{p}=\mathbf{e}pa\), \(p=0\), \(1\),..., \(N\), where \(N\) is a number of atoms. We assume the tunneling to be the predominant mechanism of interatomic coupling and neglect other ones (such as Forster and the radiation field transfer). As it was shown in [41], such an assumption can be justified for a wide range of realistic parameter values. Figure 1: **(a)** General illustration of the periodic two-level atomic chain used as a model indicating BO of electrons dressed with optical photons. It is excited with incident quantum light in the strong coupling regime. It is driven with dc voltage applied to the ends. The neighboring atoms are coupled via interatomic tunneling with different values of penetration for the ground and excited states. **(b)** Ground and excited energy levels of single two-level atom, separated by the transition energy, \(\hbar\omega_{0}\). The transition from the ground level to excited one (and vice versa) is accompanied with absorption (emission) of a single photon with energy, \(\hbar\omega\). Before starting the consideration, we will discuss the potential degree of correspondence of our model to the different types of chains. Three rather conventional simplifications have been done: (i) We neglect all types of damping; (ii) we use so called rotating-wave approximation (RWA) [26]; and (iii) we assume the atomic chain to be infinitely long and perfectly periodic. Assumption (i) relates to the case for scattering and radiation times strongly exceed the Bloch and Rabi periods. Because of RO for the quantum light are non-periodic, we imply here the value \(t_{R}\cong\Omega_{\langle n\rangle}^{-1}\), as the Rabi-period (\(\langle n\rangle\) is the mean number of photons). These conditions are met with a large margin in superconductor junctions (artificial fluxonium atoms) [53; 54; 55; 56]. The damping of BO and RO in heterostructures is defined by the electron scattering and decoherence, respectively. The recent progress in molecular beam epitaxy allowed achieving the values of scattering time \(\tau\cong 10^{-10}\) s in the ultra-high-quality AlGaAs/GaAs heterostructures [57]. For this case BO may be considered as ballistic, while dephasing becomes the dominant component of the damping for coherent inter-sub-band THz transitions (the typical values of characteristic times are 100-300 fs [50]). Such values are comparable with typical Bloch and Rabi periods; therefore, the damping does not manifest itself. In [50], the authors experimentally observed the manifestation of RO for the pulse with duration 200 fs, which does not exceed the dephasing time and is comparable with Rabi period (1-2 RO cycles over the pulse). For validity of our model to this case, we will consider interaction with rather short pulse, of which the duration, however, strongly exceeds the period of high-frequency filling and guaranties RWA validity. The perfect periodicity means the identity and a rather large number of atoms in the chain. Its implementation is rather simple for 2D-heterostructures (for example, a chain of 51 element was used in experiments [50]). The problem is not so easy for QD-chains because of QDs comprise hundreds or thousands of real atoms, with inevitable variations in size and shape and, consequently, unavoidable variability in their energies and relaxation times [58; 59]. One of the most promising technologies for applications in quantum photonics is the embedding of QDs within nanowires [59].