Evidence of dual Shapiro steps in a Josephson junctions array

The modern primary voltage standard is based on the AC Josephson effect and the ensuing Shapiro steps, where a microwave tone applied to a Josephson junction yields a constant voltage hf/2e (h is Planck's constant and e the electron charge) determined by only the microwave frequency f and fundamental constants. Duality arguments for current and voltage have long suggested the possibility of dual Shapiro steps–that a Josephson junction device could produce current steps with heights determined only on the applied frequency. In the first part we deal with the experimental results: in our setup we embed an ultrasmall Josephson junction in a high impedance array of larger junctions to reveal dual Shapiro steps. For multiple frequencies, we detect that the AC response of the circuit is synchronised with the microwave tone at frequency f, and the corresponding emergence of flat steps in the DC response with current 2ef, equal to the transport of a Cooper pair per tone period. The second part presents the theoretical tools developed to understand and simulate the complex behavior of an ultrasmall Josephson junction embedded in an array of larger junctions. We analytically and numerically study the non-linear equations of motion of a chain of junctions coupled to an electromagnetic environment. This work sheds new light on phase-charge duality, omnipresent in condensed matter physics, and extends it to Josephson circuits. Looking forward, this result opens a broad range of possibilities for new experiments in the field of circuit quantum electrodynamics and is an important step towards the long-sought closure of the quantum metrology electrical triangle.