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

TL;DR

The paper demonstrates that an overdamped RSJ model with a current-phase relation containing only two Josephson harmonics, $I_s(\varphi)=I_1\sin\varphi+I_2\sin2\varphi$, exhibits all fractional Shapiro steps under monochromatic driving. By employing perturbation theory with feedback, it derives explicit step amplitudes across weak- and strong-drive limits and shows that nontrivial fractional steps arise from the interplay of the two harmonics, with a hierarchical scaling $\Delta j_{\pm n/k}\propto j_1^{\alpha}j_2^{\beta} j_{ac}^{n}$ constrained by $\alpha+2\beta=k$. The analysis also reveals a Josephson diode effect when a phase shift between the harmonics is present, producing asymmetries between positive and negative steps. The results unify the appearance of all fractional Shapiro steps in this minimal two-harmonic CPR and provide analytical guidance on step sizes in various regimes, with implications for experiments near the tunneling limit and for systems exhibiting higher-harmonic CPRs.

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.

Figures (4)

• Figure 1: CVC of the Josephson junction with second Josephson harmonic in the CPR at $j_\mathrm{ac} = 0.8$, $j_1 = 1$, and $A = 0.7$. (a) Range of voltages below the first Shapiro step. Many fractional steps besides half-integer one are seen. (b) Range of voltages below the $1/2$ Shapiro step [zoomed part of plot (a)]. Series of steps of type $1/k$ with $k>1$ and the $2/5$ step, which is part of $2/k$ series are seen. Amplitude of the $2/5$ step is much less than amplitude of the steps of type $1/k$.
• Figure 2: Sizes $\Delta j_{n/k}$ of fractional Shapiro steps from the $\pm n/3$ series as functions of $j_\mathrm{ac}$. The amplitudes of the two Josephson harmonics are taken equal, $j_1 =j_2=1$. Values of the numerator: (a) $n=1$ and $2$; (b) $n=1$ and $4$.
• Figure 3: Ratio of the amplitudes of the $2/3$ and $1/3$ fractional Shapiro steps as a function of $A$ at $j_1 = 1/3$ and different values of $j_\mathrm{ac}$: (a) $j_\mathrm{ac}=0.5$, $1.0$, $1.5$; (b) $j_\mathrm{ac}=1.8$, $2.0$, $2.5$. Numerically calculated data points are shown by symbols (connected by straight lines for convenience). Accurate numerical determination of the step sizes becomes challenging in the case of small steps.
• Figure 4: CVC of the JJ with two Josephson harmonics in the CPR at $j_\mathrm{ac} = 2.5$, $j_1 = 1$, $A = 1$, and $\widetilde{\phi}=\pi/2$. Nontrivial value of the phase shift $\widetilde{\phi}$ between the harmonics leads to the Josephson diode effect. The figure demonstrates asymmetry between the positive and negative fractional Shapiro steps (the negative ones are present although hardly visible at the chosen values of the parameters).