# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 4 Numerical Modelling and Discussion
The processes of photons emission and absorption correspond to the rapid jumps from value \(\langle n\rangle\approx 0.1\) to \(\langle n\rangle\approx 0.9\) (and vice versa). Comparing Figure 7a with Figure 4c, one can see these jumps occurrence in a small vicinity of the stops and turns of the wavepacket. The neighboring stopping points are separated in time by half of a Bloch circle. Sufficiently far from a stopping point, the motion is characterized by rather small acceleration. It means the existence of rather weak fluctuations of photonic number, dictated by RO and taking place with Rabi-frequency \(\Omega_{0}=2g\). This scenario dramatically differs from RO in the single atom, where the acts of photons emission-absorption correspond the harmonic oscillations with Rabi-frequency. Figure 7b displays the evolution of photonic variance. As it was mentioned above, the electrons and photons in the initial state are uncorrelated. Thus, electron-photon entanglement is poor at the intervals of motion between the stopping points, whereby the variance oscillates only due to photon number fluctuations of rather small amplitude with vacuum Rabi-frequency. The processes of photons emission-absorption are accompanied the strong narrow peaks of variance. Thus, the vicinities of stopping points are characterized by the strong non-classicality of light and high degree of its  
Figure 6: Mean number of photons for light in the coherent state with \(<n>=25\). All frequencies are normalized to the frequency of quantum transition \(\omega_{(0}\), \(\Omega_{\langle n\rangle}=0.025\). Sampling frequency = 4. (**a**) The chain with the number of atoms \(N=128\), \(\omega_{B}=0.0008\), electron initial Gaussian with \(a_{0}=1\), \(b_{0}=0\), \(u_{a}=80\), \(\sigma_{a}=10\), \(k_{a}=0\), \(t_{a}/\omega_{B}=10\), \(t_{b}/\omega_{B}=1\). (**b**) Dressed single atom (Jaynes-Cummings model). entanglement with electronic sub-system. This scenario strongly differs from the case of the harmonic oscillations of variance with Rabi-frequency for the single atom [26]. The obtained results show that such manifestations of non-classicality of light, such as electron-photon entanglement and accumulated entropy become controllable by dc field via BO. As it is well known [26], the physical origin of vacuum RO is a manifestation of spontaneous emission of the single exited state in strong coupling regime. The obtained dynamics qualitatively support the conclusion of an early paper [68], that BO dramatically changes the physical picture of spontaneous emission in the atomic chains. To illustrate the mutual influence of high-frequency and low-frequency motions, we show the spatial-temporal evolution of inversion for initial condition (34) (Figure 8). One finds out of the phase oscillations of inversion with Bloch frequency. The two halves of every Bloch period correspond to the excited and ground state, respectively. Here, BO of inversion is modulated with RO. Such dynamics are similar to the shown for the vacuum initial state at Figure 4c, however, the modulation depth strongly increases due to the influence of initial entanglement (the inversion evolves over the single Rabi-circle between zero and \(\pm 1\)). In Figure 9, we show the tunnel current for initial state (34), which as well as inversion exhibits the high level of RO modulation. Comparing it with Figure 5c, one finds again the strong influence of initial electron-photon correlations. Figure 10 displays the quantum-statistical properties of light. The evolution of mean number of photons (Figure 10a) may be considered as a periodic system of step-like beatings (dashed line), modulated with high-frequency oscillations (solid curve). The value \(\langle n\rangle\) averaged with respect to the step-period is approximately equal to 0.5, which is equal to the main photon number for both states \(|\Psi_{n1,2}\rangle\) given by Equations (20) and (21).