# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 5 Some Other Potential Experimental Implementations  
### Semiconductor 2D-Heterostructures  
The next promising candidate for the experimental implementation of the model considered above is the low-dimensional semiconducting heterostructure (see Figure 2) under femtosecond inter-sub-band excitation [50]. For example, the sample consisting of 51 GaAs quantum wells separated by barriers was implemented in [50]. A coherent excitation of the sample was created by a femtosecond pulse with a center frequency resonant to the inter-sub-band transition. The resonant line at 30 THz was homogeneously broadened with coherence time 320 Fs. The amplitude of incident field was varied inside of area (5-50) kV/cm. As a result, there were observed coherent sub-picosecond RO and manipulated in a wide range by varying the strength of the coherent driving field. The measurements [50] qualitatively agreed with the predictions of the simplest model based on Maxwell-Bloch equations for non-interactive two-level systems. Conventional simplifications, such as RWA and omitting of all types of damping have been used. The simulation of the pulse driving field was done by varying the Rabi frequency \(\Omega_{R}=\Omega_{R}(t)\). Here, we add the dc field for manipulation by atomic chain via BO. For the application of our model to the potential experiments with 2D-heterostructures, it is necessary to describe the driving quantum light as a short transient normally incident to the heterostructure. The driving process in this case is described by the instantaneous coupling coefficient \(g\). The field quantization should be modified following Appendix C. The quantum properties of light in the wavepacket are dictated by the special pair of bosonic creation-annihilation operators, which allows us to rewrite the dynamic Equations (7) and (8) as:  
\[i\hbar\frac{\partial a_{p,n}}{\partial t}=\big{(}\delta\varepsilon-\textbf{E} _{dc}\cdot e\textbf{R}_{p}\big{)}a_{p,n}+t_{a}a_{p+1,n}+t_{a}^{*}a_{p-1,n}-g(t )\sqrt{n+1}b_{p,n+1}, \tag{35}\]  
\[i\hbar\frac{\partial b_{p,n+1}}{\partial t}=-\big{(}\delta\varepsilon+\textbf{ E}_{dc}\cdot e\textbf{R}_{p}\big{)}b_{p,n+1}+t_{b}b_{p+1,n+1}+t_{b}^{*}b_{p-1,n+1}-g(t )\sqrt{n+1}a_{p,n}, \tag{36}\]  
where \(g(t)=\textbf{d}_{ab}\cdot\textbf{u}_{0}(t)/\hbar\), \(\textbf{u}_{0}(t)\) is a slow envelope of the driving pulse. The driving with electromagnetic pulse leads to chain dynamics qualitatively different from the case of monochromatic light. Figures 11 and 12 depict the temporal-spatial evolution of inversion  
Figure 10: Quantum-statistical characteristics of light in the entangled initial state. Mean number of photons for the case of atom-light initial entanglement. All frequencies are normalized to the frequency of quantum transition \(\omega_{0}\). Sampling Frequency = 4. Rabi-frequency \(\Omega_{0}=0.025\), the chain with number of atoms \(N\) = 128, \(\omega_{B}=0.0008\), electron initial Gaussian with \(a_{0}=1,\ b_{0}=0,\ u_{a}=80,\ \sigma_{a}=10,\ k_{a}=0\), \(t_{a}/\omega_{B}=10,\ t_{b}/\omega_{B}=1\). (**a**) Mean number of photons. (**b**) Photonic variance. for different relations between barrier penetrations at 1S and 2S levels. Figure 11 shows the case of identical penetrations. One finds BO over the pulse duration. The oscillations are modulated with Rabi frequency and stop with the pulse disappearing at zero inversion. The behavior becomes quite different for the case of 2S penetration strongly exceed 1S one. (Figure 12). No oscillations exist before the pulse appearance (\(t<0.9\)). The reason for this is a weak value of penetration at 1S level. The switching of pulse creates both RO and BO. The RO existence is limited by the pulse duration (\(t<1.2\)), while BO continues even after the pulse switches off. It is a result of tunneling at the 2S-level on which the penetration is much higher. The charge existence at the 2S-level is a result of resonant pumping by the optical pulse. Thus, we can consider the resonant pumping as a tool of BO triggering and speak about photon-assisted BO.