Quantum dynamics of a $4\pi$-kink in Josephson junctions parallel arrays with large kinetic inductances

We present a theoretical study of the quantum dynamics of \textit{two} magnetic fluxons (MFs) trapped in Josephson junction parallel arrays (JJPAs) with large kinetic inductances. The Josephson phase distribution of two trapped MFs satisfies a topological constraint, i.e., a total variation of Josephson phases along a JJPA is $4\pi$. In such JJPAs the characteristic length of Josephson phase distribution ("the size" of MF) is drastically reduced to be less than a single cell size. Two extreme dynamic patterns will be distinguished: two weakly interacting MFs and two merged MFs, i.e., a $4\pi$-kink. Taking into account the repulsive interaction between two MFs located in the same or adjacent cells we obtain the energy band spectrum $E_{4\pi}(p)$ for a quantum $4\pi$-kink. The coherent quantum dynamics of a $4\pi$-kink demonstrates the quantum beats with the frequency and amplitude strongly deviating from ones observed for two independent MFs. In the presence of applied dc and ac bias current of frequency $f$ a weakly incoherent quantum dynamics of a $4\pi$-kink results in the Bloch oscillations and the seminal current steps with values $I^{(n)}_{4\pi}=enf$ which are two times less than ones for two independent MFs.