All fractional Shapiro steps in the RSJ model with two Josephson harmonics

TL;DR

This paper demonstrates that incorporating two Josephson harmonics in the RSJ model captures all fractional Shapiro steps and reveals asymmetries related to the Josephson diode effect.

Contribution

It shows that only two harmonics are needed to produce all fractional Shapiro steps and explores phase shift effects causing step asymmetries.

Findings

Two Josephson harmonics suffice to generate all fractional Shapiro steps.

Phase shift between harmonics causes asymmetry in positive and negative fractional steps.

Analysis of step amplitudes using perturbative methods.

Abstract

Synchronization between the internal dynamics of the superconducting phase in a Josephson junction (JJ) and an external ac signal is a fundamental physical phenomenon, manifesting as constant-voltage Shapiro steps in the current-voltage characteristic. Mathematically, this phase-locking effect is captured by the Resistively Shunted Junction (RSJ) model, an important example of a nonlinear dynamical system. The standard RSJ model considers an overdamped JJ with a sinusoidal (single-harmonic) current-phase relation (CPR) in the current-driven regime with a monochromatic ac component. While this model predicts only integer Shapiro steps, the inclusion of higher Josephson harmonics is known to generate fractional Shapiro steps. In this paper, we show that only two Josephson harmonics in the CPR are sufficient to produce all possible fractional Shapiro steps within the RSJ framework. Using perturbative methods, we analyze the amplitudes of these fractional steps. Furthermore, by introducing a phase shift between the two Josephson harmonics, we reveal an asymmetry between positive and negative fractional steps - a signature of the Josephson diode effect.