# Bloch Oscillations in the Chains of Artificial Atoms Dressed with Photons
## 5 Some Other Potential Experimental Implementations
### Josephson Junction  
The next promising candidate for experiment implementation is a single Josephson junction embedded in an inductive environment. As it was shown in [53; 54; 55; 56], the voltage-biased Josephson junction exhibits the temporal dynamics equal to the motion of a fictitious charged particle with inertia provided by the inductance. This particle moves in a potential, which also has a periodic part and a linear tilt. On the other hand, a Josephson junction strongly coupled to the microwave resonator (Figure 3) in the resonant regime may be considered as an artificial two-level atom (\"fluxonium\" [54]). It is described by the Jaynes-Cummings model [54] and exhibits RO dynamics [54]. Here, we will focus on the case of Josephson junction simultaneously driven by dc voltage and quantum light in the regime of small detuning. We will show that the model investigated in this work is applicable to this type of physical system. Therefore, we will start our analysis by identifying the parameters of Josephson junction in terms of this model. The temporal dynamics of entire circuit may be described using so-called quasi-charge representation [53; 55]. Let us first consider the single junction, of which the standard form reads \(\hat{H}_{0}=\left(\hat{Q}^{2}/2C_{J}\right)-E_{J}\cos\phi_{J}\), where \(\hat{Q}=(2e/i)\partial/\partial\phi_{J}\), \(\hat{\phi}_{J}\) are operators of junction charge and phase difference across the junction, respectively. It describes the particle of mass \(C_{J}\) moving in the periodic potential [55] (\(C_{J}\) is a junction capacitance). The eigenfunctions of this Hamiltonian are Floquet-Bloch modes with condition of quasi-periodicity \(\Psi_{\alpha,Q}(\varphi)=P_{\alpha,Q}(\varphi)e^{i\varphi Q/2e}\), where \(P_{\alpha,Q}(\varphi)=P_{\alpha,Q}(\varphi+2\pi)\) is periodic function, \(\alpha\) is a number of the gap. These functions describe the motion of the quasiparticles, which is similar to the motion of electrons in crystal. The phase \(\varphi\) and quasi-charge \(Q\) play the role of the conjugated spatial coordinate and quasi-momentum for electrons in crystals, respectively. The energies of the two first modes \(\varepsilon_{0,1}(Q)\), are separated by the gap, which in the regime of interest here (\(E_{J}>>E_{C_{J}}\), \(E_{C_{J}}=e^{2}/2C_{J}\)) is of the order of plasma frequency \(\omega_{p}=\sqrt{8E_{I}E_{C_{J}}}/\hbar\). Taking \(E_{J}/h=30\) GHz, and \(E_{C_{J}}/h=3\), we have \(\omega_{p}=150\) GHz. Next, in the limit \(E_{J}>>E_{C_{J}}\) the energy bands \(\varepsilon_{0,1}(Q)\) are purely sinusoidal [53] with the bandwidths of the order of [55]  
\[\Delta_{0}=16\sqrt{E_{C_{J}}E_{J}/\pi}\Big{(}E_{J}/2E_{C_{J}}\Big{)}^{1/4}e^{- \sqrt{8E_{J}/E_{C_{J}}}} \tag{37}\]  
(the typical values are \(\Delta_{0}\approx E_{C}/2=1,5\) GHz). The situation is the same in the case of atomic chain considered above (see Equation (13) in the case \(g_{n}=0\)). Thus, we are able to establish the correspondence \(2t_{a,b}\leftrightarrow\Delta_{0}\). The additional series inductance and switching on the bias voltage transforms the Hamiltonian \(\hat{H}_{0}\) to  
\[\hat{H}_{A}=\hat{Q}^{2}/2C_{J}+E_{L_{J}}(\phi_{b}-\hat{\phi}_{J})^{2}/2+E_{J} \cos(\hat{\phi}_{J})\,, \tag{38}\]  
where \(E_{L_{J}}=(\hbar/2e)^{2}L_{2}^{-1}\) is the constant of inductance energy, \(\phi_{b}\) is the total phase difference across the circuit, which is determined by the bias voltage \(V_{b}\), such that \(\dot{\phi}_{b}=2 eV_{b}/\hbar\)[53]. The case under consideration has a simple physical interpretation based on the charge-phase duality [53]. The voltage-biased junction with inductive environment and the current-biased junction shunted by a capacitor \(C_{J}\) are dual to each other when exchanging the role of quasi-charge and phase. The dynamics of the circuit, shown in Figure 3, are equal to the intra-band motion of a fictitious charged particle in the tilted periodic potential.