Fractional Shapiro steps in ladder Josephson arrays

We study the response of Josephson ladder arrays (1×N plaquettes) to a combined ac and dc current. For N≥1, the arrays are predicted to exhibit Shapiro steps at voltages 〈V〉=nNħω/(2e) for n=1/2,1,3/2, . . ., over a broad range of fields f=Φ/${\mathrm{\Phi }}_0$ (Φ=flux per plaquette, ${\mathrm{\Phi }}_0$=hc/2e). The calculated half-steps persist even in single-plaquette ‘‘arrays’’ at all fields f≠0, and, when the critical currents parallel to the applied current are sufficiently unequal, even at f=0. Measurements of the step widths Δ${\mathit{I}}_{1/2}$(f) and Δ${\mathit{I}}_1$(f) for 1×600 ladder arrays of Nb-Au-Nb junctions are in good agreement with the calculations.