Josephson junction dynamics in the presence of $2\pi$- and $4\pi$-periodic supercurrents

We investigate theoretically the dynamics of a Josephson junction in the framework of the resistively shunted junction model. We consider a junction that hosts two supercurrent contributions: a $2\pi$ and a $4\pi$ periodic in phase, with intensities $I_{2\pi }$ and $I_{4\pi }$, respectively. We study the size of the Shapiro steps as a function of the ratio of the intensity of the mentioned contributions, i.e., $I_{4\pi }/I_{2\pi }$. We provide detailed explanations where to expect clear signatures of the presence of the $4\pi$-periodic contribution as a function of the external parameters: the intensity ac bias $I_{\text{ac}}$ and frequency ${\omega }_{\text{ac}}$. On the one hand, in the low ac-intensity regime (where $I_{\text{ac}}$ is much smaller than the critical current $I_{\text{c}}$), we find that the nonlinear dynamics of the junction allows the observation of only even Shapiro steps even in the unfavorable situation where $I_{4\pi }/I_{2\pi }\ll 1$. On the other hand, in the opposite limit ($I_{\text{ac}}\gg I_{\text{c}}$), even and odd Shapiro steps are present. Nevertheless, even in this regime, we find signatures of the $4\pi$ supercurrent in the beating pattern of the even step sizes as a function of $I_{\text{ac}}$.

Figure 1

(a) Scheme of the RSJ circuit. (b) $\overline{V}$ as a function of $I_0$, with $I_{\text{ac}}=0$. The voltage becomes finite for $I_0\ge I_{\text{c}}$. The dependency is $\overline{V}\sim R\sqrt{I_0^2-I_{\text{c}}^2}$. (c), (d) We represent the voltage as a function of $I_0$, with $I_{\text{ac}}\ne 0$, and $I_{4\pi }=0$ (c), and $I_{2\pi }=0$ (d). Thus, the periodicity of the supercurrent is reflected in the parity of the Shapiro steps. In panel (c) [(d)], we show Shapiro steps at integer (even) multiples of $\hslash {\omega }_{\text{ac}}/2e$.

Figure 2

$I\text{−}\overline{V}$ curves for different values of $I_{\text{ac}}=0$ up to $I_{\text{c}}$, with $I_{4\pi }/I_{2\pi }=0.5,{\omega }_{\text{ac}}=0.2(2eR{I}_{\text{c}}/\hslash )$. We observe the appearance of odd steps, when $I_{\text{ac}}\gtrsim I_{4\pi }$.

Figure 3

(a) The washboard potential with $I_{\text{ac}}=0$ as a function of $\phi$ for three different values of $I_0$: top curve $I_0=0,I_0<I_{\text{c}}$ middle curve, and $I_0>I_{\text{c}}$ bottom curve. The dashed lines remark the slope of the WP at the odd ($F_1$) and even ($F_2$) flattest regions. We highlight the even and odd sectors in red and blue, respectively. (b) We show the time evolution of $\dot{\phi }\left(\tau \right)$ [31]. We mark in blue (red) the odd (even) sectors according to Eqs. (3) and (4). Besides, we can extract from the WP the maxima of $\dot{\phi }\left(t\right):S_1\approx I_0+I_{\text{c}}-\sqrt{2}{I}_{4\pi },S_2=I_0+I_{\text{c}},F_1=I_0-I_{\text{c}}$, and $F_2\approx I_0-I_{\text{c}}+\sqrt{2}{I}_{4\pi }$, being the steepest and the flattest slopes in each sector, and the equation $(\hslash /2eR)\dot{\phi }=-\partial U(\phi ,t)/\partial \phi$ relates the slope and the velocity at each time. (c) We represent the ratio $T_1/T_2$ as a function of $I_0$ for different values of $I_{4\pi }/I_{2\pi }$ from zero to one.

Figure 4

The renormalized washboard potential $\tilde{U}(\phi ,t)$ as a function of $\phi$ for two different values of the external bias $I_0-I_{\text{c}}<I_{\text{ac}}<I_0-I_{\text{c}}+\sqrt{2}{I}_{4\pi }$ (bottom curve) and $I_0-I_{\text{c}}+\sqrt{2}{I}_{4\pi }<I_{\text{ac}}$ (top curve). We highlight in green the sectors where $\partial \tilde{U}/\partial \phi <0$ and in red $\partial \tilde{U}/\partial \phi >0$.

Figure 5

Phase diagram of the voltage as a function of $I_0$ and $I_{\text{ac}}$. We differentiate between “no motion regime” (red area), where $\overline{V}=0$. The “linear regime” (yellow area), where there is no Shapiro steps but $\overline{V}\ne 0$. Finally, the “Shapiro steps regime” (green and blue areas). Remarkably, following the WP considerations we expect to observe only even steps in the blue area, i.e., for $I_0-I_{\text{c}}+\sqrt{2}{I}_{4\pi }\gtrsim I_{\text{ac}}\gtrsim I_0-I_{\text{c}}$.

Figure 6

Low-intensity limit $I_{\text{ac}}\ll 1$: Fourier transform (color scale) ${\dot{\phi }}_0\left(\omega \right)=|\int dt{e}^{i\omega t}{\dot{\phi }}_0\left(t\right)|$ as a function of $\omega$ and the voltage $V={\omega }_0$. The intensity of the resonances follow Eq. (9). The first two resonance lines with higher slope correspond to the frequencies $\omega ={\omega }_0/2$ (fractional frequency) and $\omega ={\omega }_0$. The rest of the resonance lines correspond to higher harmonics.

Figure 7

Low-intensity limit $I_{\text{ac}}\ll 1$: first (dashed curve) and second (solid curve) Shapiro steps width in units of $I_{\text{ac}}$ as a function of ${\omega }_{\text{ac}}$. The width of the Shapiro steps is calculated from Eq. (15).

Figure 8

High-intensity limit $I_{\text{ac}}\gg 1$: width of the first two steps as a function of $I_{\text{ac}}/{\omega }_{\text{ac}}$. The oscillatory behavior is due to the Bessel functions [see Eqs. (21) and (22)]. Interestingly, even Shapiro steps exhibit a beating pattern produced by the coexistence of $2\pi$ and $4\pi$ supercurrents.

Figure 9

Comparison between the numerical solution of Eq. (5) ${\dot{\phi }}_0\left(\tau \right)$ (solid lines) and the approximate solution given by Eq. (8) (dashed lines). We have used $I_{4\pi }/I_{2\pi }=0.5$. We compare two different values of $I_{\text{v}}=1.1$ (left panel) and $I_{\text{v}}=2.1$ (right panel).