Bloch oscillation and topological quantization

We study dynamic responses of two-dimensional arrays of ideal Josephson junctions at zero temperature. In the appropriate limit the dc component of the applied current is shown to be quantized, which is a generalization of the Bloch oscillation previously considered in a single Josephson junction. The vortex description manifests the topological character of the quantization. The possibility of fractional quantization associated with the ground-state degeneracy is also pointed out. This together with the giant Shapiro steps and the quantum Hall effect suggests the three types of topological quantization present in the system.