# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 4 Numerical Modelling and Discussion
In Figure 5, we plotted the time dependence of the tunneling current. For the absence of light (Figure 5a), the current exhibits the periodic dynamics with Bloch frequency. This picture is dramatically changed due to the light-chain coupling both for coherent and vacuum states of light (Figure 5b,c), respectively. For the coherent light, the tunnel current is modulated in exact compliance with the collapse-revivals picture. This result is rather counterintuitive; it demonstrates the influence of the quantum-light statistics on the low-frequency motion of the charges driven separately by a dc field. It makes the spectra of the tunneling current more wide and various wherewith in the case of ordinary BO. It opens a new promising avenue in the spectroscopy of nano-circuits and nano-devices based on the synthesis of quantum optical and dc tools of control [51; 52]. For the vacuum state of light, the tunnel current periodically evolves through one pulse after another with frequency \(\omega_{B}\) instead of continuous periodic behavior in the case of ordinary BO [8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25]. Such evolution is accompanied by the modulation with Rabi frequency. The results demonstrate high and various mutual interactions of RO and BO. Figure 4: Space-time distribution of inversion density. All frequencies are normalized to the frequency of quantum transition \(\omega_{0}\). Sampling Frequency = 4. The number of atoms \(N\) = 128, Rabi-frequency \(\Omega_{\langle n\rangle}=0.025\), electron initial Gaussian with \(a_{0}=1,\ b_{0}=0\), \(u_{a}=80\), \(\sigma_{a}=10\), \(k_{a}=0\). (**a**) Light in the coherent state with \(<n>=25\). \(\omega_{B}=0\), \(t_{a}/\omega_{B}=t_{b}/\omega_{B}=10\). (**b**) Light in the coherent state with \(<n>=25\). \(\omega_{B}=0.0008\), \(t_{a}/\omega_{B}=10\), \(t_{b}/\omega_{B}=1\). (**c**) light in vacuum. \(\omega_{B}=0.0008\), \(t_{a}/\omega_{B}=10\), \(t_{b}/\omega_{B}=1\). Figure 5: Space-time distribution of tunnel current density. All frequencies are normalized to the frequency of quantum transition \(\omega_{0}\). Sampling Frequency = 4. The number of atoms \(N=128\), Rabi-frequency \(\Omega_{\langle n\rangle}=0.025\), electron initial Gaussian with \(a_{0}=1,\ b_{0}=0\), \(u_{a}=80\), \(\sigma_{a}=10,\ k_{a}=0\). (**a**) Light in the coherent state with \(<n>=25\). \(\omega_{B}=0\), \(t_{a}/\omega_{B}=t_{b}/\omega_{B}=1\). (**b**) Light in the coherent state with \(<n>=25\). \(\omega_{B}=0.0008\), \(t_{a}/\omega_{B}=10\), \(t_{b}/\omega_{B}=1\). (**c**) Light in vacuum state. \(\omega_{B}=0.0008\), \(t_{a}/\omega_{B}=10\), \(t_{b}/\omega_{B}=1\). In Figure 6, we plot the dynamics of mean number of photons. Again, we see the mutual influence of RO and BO. For dc field absence (Figure 6b), the collapse-revivals scenario corresponds the Jaynes-Cummings model for single atom [26]. It represents the quantum interference of the spectral components with different Rabi-frequencies \(\Omega_{n}\) and different relative amplitudes [26]. For dc field appearance (Figure 6a), there is one more type of interference components with frequencies \(m\omega_{B}\) (\(m\) is integer value). Among them, the term with \(m=1\) is dominant, and its amplitude is comparable with RO components. As a result, the mean number of photons is modulated with Bloch frequency \(\omega_{B}\). Therefore, the dc field via BO opens one more way of engineering of the quantum light states. This effect depends on the large number of physical factors, such as dc field value, energy and dipole moment of quantum transition for the atom, tunnel coupling, etc. Concluding, the results of numerical modeling agree with the simple analytical model, developed in Section 3. They are promising for different applications in quantum computing, quantum informatics, etc. Figure 7 shows the quantum-statistical properties of light. The temporal behavior of the mean number of photons (Figure 7a) may be rather simply understood from the quasi-classical point of view developed in Section 3.