# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 4 Numerical Modelling and Discussion  
In this section, we will show and discuss the numerical results for the different types of initial states. The System (8) and (9) was solved with the Crank-Nicolson numerical integration technique [67]. We have used periodic Born-Von Karman relations \(a_{1,n}=a_{N,n},b_{1,n}=b_{N,n}\) as boundary conditions [41]. Let us note that the concrete form of boundary conditions has no physical importance in our case because the area of oscillations is placed rather distantly from the ends of the chain. For simplicity, we will limit our consideration by the case of zero detuning (\(\omega=\omega_{0}\)). We will consider three types of initial conditions. For two of them the electron and photon subsystems are assumed to be initially non-interacting. The electronic subsystem is prepared as a coherent superposition of Gaussian wave packets:  
\[a_{p}(0)=a_{0}e^{-\frac{(p-a_{0})^{2},2}{\sigma_{0}^{2}}}e^{ik_{k}pa}, \tag{30}\]  
\[b_{p}(0)=b_{0}e^{-\frac{(p-a_{0})^{2},2}{\sigma_{0}^{2}}}e^{ik_{k}pa}, \tag{31}\]  
where \(a_{0},b_{0}\) are arbitrary complex values that satisfy the normalization condition, \(u_{a,b},\sigma_{a,b},k_{a,b}\) are the position of Gaussian center, effective Gaussian width, and the initial value of quasi-momentum at the ground and excited states, respectively. The photonic subsystem is prepared in the next types of state:  
(i) Coherent initial state; It is given by Poisson distribution \(p(n)=\langle n\rangle^{n/2}e^{-\langle n\rangle/2}/\sqrt{n! }\), where \(\langle n\rangle\) is the mean number of photons. The total wavefunction at the moment \(t=0\) is given by  
\[\ket{\psi(0)}=\sum_{n}\frac{\langle n\rangle^{n/2}}{\sqrt{n!}}e^{-\frac{ \langle n\rangle}{2}}\sum_{p}\left(a_{0}e^{-\frac{(p-u_{p})^{2}a^{2}}{\sigma_ {a}^{2}}}e^{ik_{a}pu}\big{|}a_{p},n\rangle+b_{0}e^{-\frac{(p-u_{p})^{2}a^{2}}{ \sigma_{b}^{2}}}e^{ik_{b}pu}\big{|}b_{p},n\rangle\right); \tag{32}\]  
(ii) _Vacuum initial state_; It is given by the total wave function  
\[\ket{\psi(0)}=\sum_{p}\left(a_{0}e^{-\frac{(p-u_{p})^{2}a^{2}}{\sigma_{a}^{2}}} e^{ik_{a}pu}\big{|}a_{p},0\rangle+b_{0}e^{-\frac{(p-u_{p})^{2}a^{2}}{\sigma_{b}^{2} }}e^{ik_{b}pu}\big{|}b_{p},0\rangle\right), \tag{33}\]  
where \(a_{0},b_{0}\) are the arbitrary coefficients satisfying normalization condition;  
(iii) _Entangled photon-electron initial state_; The electron-photon entanglement prepared initially before the driving fields would be switching on. The initial condition reads  
\[\ket{\psi_{n}(0)}=C_{0}\sum_{p}e^{-\frac{(p-u)^{2}a^{2}}{\sigma^{2}}}\big{(} \big{|}a_{p},n\rangle+\big{|}b_{p},n+1\rangle\big{)}, \tag{34}\]  
where \(C_{0}\) is normalization constant. The value of inversion for the State (34) is equal to zero. In Figure 4, we plotted the temporal dynamics of inversion. The case of zero dc field and coherent initial light (Figure 4a) corresponds to the well-known scenario of collapse-revivals given by the Jaynes-Cummings model for the single atom [26]. The initial spatial distribution is not disturbed over the motion via inter-atomic tunneling. For the case of dc field existence and coherent initial light (Figure 4b), the collapse-revivals picture drifts due to BO: The collapses and revivals appear at the different segments of the atomic chain. The inversion behavior dramatically changes for the vacuum state of light (Figure 4c). Here, BO leads to the separation of areas of positive and negative values of inversion. The particle in the excited state (positive inversion) exhibits the single oscillation of rather high amplitude in the course a half of Bloch circle. During the other part of Bloch circle, the particle is in rest, which means its appearance in the ground state (negative inversion). The reason for this is the weak value of transparence of potential barrier at the ground level used in the shown calculations. In contrast with single atom scenario, vacuum RO between maximal and minimal values of inversion occurs with Bloch frequency \(\omega_{B}\) instead of Rabi frequency \(\Omega_{0}=2g\).