Switching times in long-overlap Josephson junctions subject to thermal fluctuations and non-Gaussian noise sources

We investigate the superconducting lifetime of long current-biased Josephson junctions, in the presence of Gaussian and non-Gaussian noise sources. In particular, we analyze the dynamics of a Josephson junction as a function of the noise signal intensity, for different values of the parameters of the system and external driving currents. We find that the mean lifetime of the superconductive state is characterized by nonmonotonic behavior as a function of noise intensity, driving frequency, and junction length. We observe that these nonmonotonic behaviors are connected with the dynamics of the junction phase string during the switching towards the resistive state. An important role is played by the formation and propagation of solitons, with two different dynamical regimes characterizing the dynamics of the phase string. Our analysis allows to evidence the effects of different bias current densities, that is a simple spatially homogeneous distribution and a more realistic inhomogeneous distribution with high current values at the edges. Stochastic resonant activation, noise-enhanced stability, and temporary trapping phenomena are observed in the system investigated.

Figure 1

(a) Washboard potential at three different times with a soliton wave ($2\pi$ kink) on the highest profile, (b) soliton [Eq. (9)] and corresponding fluxon profile [Eq. (10)], and (c) circuit diagram of a JJ noise detector: a JJ with critical current $i_C$ is biased in a twofold way.

Figure 2

Inhomogeneous bias current density [see Eq. (7)] along JJs, for $i_0=0.9$ and different values of junction length.

Figure 3

Probability density functions for Gaussian (solid line), Cauchy-Lorentz (dashed line), and Lévy-Smirnov (dash-dotted line) distributions.

Figure 4

MST $\tau$ vs $L$ for different current distributions along the junction: homogeneous $i_b\left(x\right)$ and noise sources with (a) Gaussian, (b) Cauchy-Lorentz, and (c) Lévy-Smirnov statistics; inhomogenous $i_b\left(x\right)$ and noise sources with (d) Gaussian, (e) Cauchy-Lorentz, and (f) Lévy-Smirnov statistics. In all graphs the other parameters are $i_0=\{0.5\text{(open}\text{symbols)},0.9\text{(solid}\text{symbols)}\}$, $\omega =\{0.4\text{(circles)},0.7\text{(triangles)},0.9\text{(squares)}\}$, and $\gamma =0.2$. The legend in (c) refers to all figures.

Figure 5

String dynamics during the switching towards the resistive state, for a JJ of length $L=2$, with (a) homogeneous bias current distribution and G noise source, and (b) inhomogeneous bias current distribution and CL noise source, and for a JJ of length $L=15$, with (c) homogeneous bias current distribution and CL noise source, and (d) inhomogeneous bias current distribution and G noise source. All graphs were obtained for $\omega =0.9$ and $\gamma =0.2$. The curves in panels (b) and (c) show the characteristic Lévy flights of the CL statistic.

Figure 6

Log-log plots of MST $\tau$ vs $\omega$ obtained using homogeneous $i_b\left(x\right)$ and noise sources (a) G, (c) CL, and (e) LS, and inhomogeneous $i_b\left(x\right)$ and noise sources (b) G, (d) CL, and (f) LS. In all graphs the values of the other parameters are $i_0=0.9$, $L=10$, and $\gamma =\{0.025,0.1,0.2,0.45,0.9\}$. The legend in (d) refers to all panels.

Figure 7

Log-log plots of MST $\tau$ vs $\gamma$ obtained using homogeneous $i_b\left(x\right)$ and noise sources (a) G, (c) CL, and (e) LS, and inhomogeneous $i_b\left(x\right)$ and noise sources (b) G, (d) CL, and (f) LS. In all graphs the values of other parameters are $i_0=\{0.5,0.9\}$, $\omega =0.9$, and $L=\{1,10\}$. The legend in (c) refers to all panels.

Figure 8

Time evolution of the probability $P\left(t\right)$ in the following conditions: G noise with (a) $L=1$ and (b) $L=10$; (c) CL noise with $L=10$. The system parameters are $i_0=0.9$ and $\omega =0.9$. Each graph contains curves of $P\left(t\right)$ obtained using values of $\gamma$ for which a minimum or maximum appears in the $\tau$ vs $\gamma$ behavior. The insets reproduce the corresponding curves of Figs. 7 and 7.

Figure 9

MST $\tau$ as a function of $L$, ${\gamma }_{\mathrm{CL}}$, and $\omega$. All curves were obtained considering the simultaneous presence of CL and thermal noise sources, using two different values, i.e., (a, c, e) $i_0=0.5$ and (b, d, f) $i_0=0.9$, of the homogenous bias current, and varying the Gaussian noise intensity, ${\gamma }_{\mathrm{G}}$. The legend in (d) refers to all panels.