Phase Retrapping in a Pointlike $\phi$ Josephson Junction: The Butterfly Effect

We consider a $\phi$ Josephson junction, which has a bistable zero-voltage state with the stationary phases $\psi =\pm \phi$. In the nonzero voltage state the phase “moves” viscously along a tilted periodic double-well potential. When the tilting is reduced quasistatically, the phase is retrapped in one of the potential wells. We study the viscous phase dynamics to determine in which well ($-\phi$ or $+\phi$) the phase is retrapped for a given damping, when the junction returns from the finite-voltage state back to the zero-voltage state. In the limit of low damping, the $\phi$ Josephson junction exhibits a butterfly effect—extreme sensitivity of the destination well on damping. This leads to an impossibility to predict the destination well.

Figure 1

Tilted periodic double-well (Josephson) potential $U(\psi )$ given by Eq. (3) for ${\Gamma }_0=-4$ and different values of bias current (tilt) $\gamma$.

Figure 2

The dependence ${\gamma }_R(\alpha )$ for different values of ${\Gamma }_0$. Symbols represent the results of direct numerical simulation. Lines show PT results given by Eq. (10) for $\alpha \to 0$. The horizontal dashed lines show the values of the depinning current ${\gamma }_{c-}$ for given ${\Gamma }_0$, i.e., the current, at which the $-\phi$ well disappears. For $\gamma >{\gamma }_{c-}$ the potential has only one $+\phi$ well and the phase is retrapped there. The vertical dashed line shows the corresponding value of ${\alpha }_R({\gamma }_{c-})$. For JJ with $\alpha >{\alpha }_R({\gamma }_{c-})$ the retrapping current ${\gamma }_R(\alpha )>{\gamma }_{c-}$ and potential has only one $+\phi$ well where the phase is retrapped.

Figure 3

The retrapping trajectories in the phase plane ($\dot{\psi }$, $\psi$) for ${\Gamma }_0=-3$ and different tilt $\gamma$. The phase starts at ${\psi }_L$, where $U(\psi )$ has a maximum, with $\dot{\psi }=0$ and arrives to ${\psi }_R={\psi }_L+2\pi$ [the next $U(\psi )$ maximum] with $\dot{\psi }=0$. Then the phase falls back and is trapped in one of the minima of the $U(\psi )$.

Figure 4

The well in which the particle is trapped at ${\gamma }_R(\alpha )$ as a function of $\alpha$ (symbols). Vertical dashed line show the value ${\alpha }_R({\gamma }_{c-})$. For $\alpha >{\alpha }_R({\gamma }_{c-})$ the potential has only a single $+\phi$ well. In (a) ${\alpha }_R({\gamma }_{c-})\approx 1.327$ and is not visible. The lines show the effect of the low frequency Gaussian noise with ${\sigma }_{\gamma }=0.02$ in the bias current circuitry. In this case the right vertical axis represents the probability $P_{-}$ to find the system in the $-\phi$ state.

Figure 5

The dependence $N(\alpha )$ for different values of ${\Gamma }_0$. At low $N$ one sees that $N$ is an integer. At high $N$ it is a straight line on this double log scale, i.e., it corresponds to $N\propto 1/\alpha$.