Topological Quantization in Superconducting Arrays

Fractional giant Shapiro resonances in frustrated systems driven by a time-dependent flux

“…Here frustration changes the periodicity of the persistent current from the fundamental period to the fractional one, depending on the commensurability of the number of lattice sites [4]. Frustration also yields interesting effects on dynamic responses: For example, planar arrays of superconductors, weakly coupled by Josephson junctions, display quantized voltage plateaus called giant Shapiro steps in the presence of external driving currents; frustration here produces fractional steps in addition to the usual integer steps [5], [6]. Unlike the planar system, which may be driven by currents, a system with non-simply-connected geometry can also be driven by a time-dependent threading flux [7].…”

“…The voltage plateaus, namely, the giant Shapiro steps, displayed by current-driven arrays have been pointed out to be topological quantization [6]. In a similar manner, the topological origin of the flux quantization leading to integer and fractional giant Shapiro resonances in flux-driven systems can be manifested in the following way: The threading flux φ = φ 0 + φ d t + φ a cos ωt is periodic in time with period mτ ≡ 2πm/ω only if φ d = (n/m)(ω/2π) since the dc component has the periodicity 1/φ d .…”