Modeling Bloch oscillations in nanoscale Josephson junctions

Bloch oscillations in nanoscale Josephson junctions with a Coulomb charging energy comparable to the Josephson coupling energy are explored within the context of a model previously considered by Geigenm\"uller and Sch\"on that includes Zener tunneling and treats quasiparticle tunneling as an explicit shot-noise process. The dynamics of the junction quasicharge are investigated numerically using both Monte Carlo and ensemble approaches to calculate voltage-current characteristics in the presence of microwaves. We examine in detail the origin of harmonic and subharmonic Bloch steps at dc biases $I=(n/m)2ef$ induced by microwaves of frequency f and consider the optimum parameters for the observation of harmonic ($m=1$) steps. We also demonstrate that the GS model allows a detailed semiquantitative fit to experimental voltage-current characteristics previously obtained at the Chalmers University of Technology, confirming and strengthening the interpretation of the observed microwave-induced steps in terms of Bloch oscillations.