Quantum dynamics of a single fluxon in Josephson junctions parallel arrays with large kinetic inductances

We present a theoretical study of coherent quantum dynamics of a single magnetic fluxon (MF) trapped in Josephson junction parallel arrays (JJPAs) with large kinetic inductances. The MF is the topological excitation carrying one quantum of magnetic flux, $\Phi_0$. The MF is quantitatively described as the $2\pi$-kink in the distribution of Josephson phases, and for JJPAs with high kinetic inductances the characteristic length of such distribution ("the size" of MF) is drastically reduced. Characterizing such MFs by the Josephson phases of three consecutive Josephson junctions we analyse the various coherent macroscopic quantum effects in the MF quantum dynamics. In particular, we obtain the MF energy band originating from the coherent quantum tunnelling of a single MF between adjacent cells of JJPAs. The dependencies of the band width $\Delta$ on the Josephson coupling energy $E_J$, charging energy $E_C$ and the inductive energy of a cell $E_L$, are studied in detail. In long linear JJPAs the coherent quantum dynamics of MF demonstrates decaying quantum oscillations with characteristic frequency $f_{qb}=\Delta/h$. In short annular JJPAs the coherent quantum dynamics of MF displays complex oscillations controlled by the Aharonov-Casher phase $\chi \propto V_g$, where $V_g$ is an externally applied gate voltage. In the presence of externally applied dc bias, $I$, a weakly incoherent dynamics of quantum MF is realized in the form of macroscopic Bloch oscillations leading to a typical "nose" current-voltage characteristics of JJPAs. As ac current with frequency $f$ is applied the current-voltage characteristics displays a set of equidistant current steps at $I_n=2en f$.