# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 2 Statement of the Problem and Calculation Technique
### Physical System and Model
The QDs form at the apex of a GaAs/AlGaAs interface, are highly stable, and can be positioned with nanometer precision relative to the nanowire centre. As it was found [59], there is a chain of bright, nanoscale emitters in the red. QD-in-nanowire mimics very closely a two-level atom with high associated lifetime (\(\cong\)450 ps). One more way is using a scanning tunneling microscope to create QDs with identical, deterministic sizes [58]. As it was mentioned in [58], the reached digital fidelity opens the door to QD architectures free of intrinsic broadening. This makes it reasonable to recommend our model of 1D QD-chain for visible light applications. There exist different ways to theoretically describe complex quantum systems strongly interacted with quantum light. In quantum optics an \"all-matter\" picture is widely used, where the dynamics of light is integrated out (for example, in the optical Bloch equations [60; 61]). In Refs. [62; 63] the light-matter interaction is treated in an \"all-light\" picture (Lippmann-Schwinger equation approach). We use as the theoretical approach the probability amplitude method, generalized for the case of 1D chains driven via homogeneous dc field. It consists of solving of the Schrodinger equation for wave function, which is the superposition of various Wannier-Fock states for atom-light system. To solve equations of motion for probability amplitudes analytically, we use the quasi-classical concept. The dressed electron is described as a wavepacket prepared with a well-defined quasi-momentum. The motion of quasiparticle center of mass is governed by the [PERSON]'s law, while the internal degrees of freedom have been described by means of quantum theory, based on the concept of electron-photon entanglement widely used in quantum optics [26; 27]. Thus, the position of the quasi-particle center periodically evolves with a Bloch frequency corresponding to the quasi-momentum scanning a complete Brillouin zone. Such an approach was used for BO of conductive electrons in [8; 64] and will be adopted for the case of dressed electrons in this paper. In the general case, the equations of motion were integrated numerically with simplification through RWA. However, the use of the RWA may not describe the atom-light interaction when the coupling becomes sufficiently strong (ultrastrong coupling) [65]. Both lower and upper fundamental limitations of ultrastrong coupling of atoms and light were recently formulated in [66]. We bounded our consideration by the strong coupling regime assuming the lower limitation in [66] not to be reached. Figure 3: Circuit diagram of a Josephson junction (crossed), which is inductively coupled with microwave resonator. The inductance \(L_{2}\) describes the inductive environment of the junction. The inductance \(L_{1}\) is a tool of control of the \"junction-resonator\" coupling (it is assumed \(L_{2}>>L_{1}\)). Josephson energy \(E_{J}\) strongly exceeds charging energy \(E_{C}\) (\(E_{J}>>E_{C}\)). We model the resonator as an LC-circuit with effective inductance \(L_{r}\) and effective capacitance \(C_{r}\). The single fluxonium plays role of atomic chain due to the interaction with inductive environment [53; 54; 55; 56] (BO take place in the space of quasi-charge). Figure 2: Schematic diagram of 2D semiconductor heterostructure. (**a**) \(n\)-type modulation-doped quantum well sample consisting from GaAs wells considered as artificial atoms forming a chain. The atoms are separated from each other by AlGaAs barriers. The coupling is governed by the inter-barrier electron tunneling. The dc voltage is applied to the ends of the system. The femtosecond THz pulse comes in through nanoantenna. (**b**) \(1\)S \(\Leftrightarrow\) 2S inter-sub-band transition, which is coherently excited by the pulse with narrow spectrum (central frequency is resonant with quantum transition).