Theory of fractional vortex escape in a long Josephson junction

We consider a fractional Josephson vortex in an infinitely long $0\text{−}\kappa$ Josephson junction. A uniform bias current applied to the junction exerts a Lorentz force acting on a vortex. When the bias current becomes equal to the critical (or depinning) current, the Lorentz force tears away an integer fluxon and the junction switches to the resistive state. In the presence of thermal and quantum fluctuations this escape process takes place with finite probability already at subcritical values of the bias current. We analyze the escape of a fractional vortex by mapping the Josephson phase dynamics to the dynamics of a single particle in a metastable potential and derive the effective parameters of this potential. This allows us to predict the behavior of the escape rate as a function of the topological charge of the vortex.

Figure 1

(a) Phase profiles $\mu \left(x\right)$ and (b) magnetic field profiles ${\mu }_x\left(x\right)$ corresponding to the static solutions (vortex with a topological charge $\wp$ pinned at the phase discontinuity at $x=0$) of the sine-Gordon equation (2) with $\wp =\pi$ and $\gamma =0$ (gray), and $\gamma =0.6$ (black). Note, that ${\gamma }_c\left(\pi \right)=2/\pi \approx 0.635$, see Eq. (1). (c) Shows the depinning dynamics, i.e., the magnetic field profile ${\mu }_x\left(x\right)$ as a function of time after the bias current $\gamma$ was abruptly increased from 0.63 (still static solution) to 0.64 (no static solution) at $t=0$. One can see a fluxon separating and moving to the left and an antisemifluxon left at the discontinuity.

Figure 2

(Color online) The frequencies ${\omega }_0^{\text{cr}}$, ${\omega }_0^{\text{pt}}$, and ${\omega }_0^{\text{nu}}$ as a function of the bias current $\gamma$ for different values of $\wp$. The gray area indicates the plasma band, which was drawn by filling up the area above ${\omega }_1^{\text{nu}}\left(\gamma \right)$.

Figure 3

(Color online) The energy barriers $\Delta V^{\text{cr}}$, $\Delta V^{\text{pt}}$, and $\Delta V^{\text{nu}}$ as a function of the bias current $\gamma$ for different values of $\wp$. Note that $\Delta V^{\text{cr}}$ diverges for $\wp \to 0$. Therefore, we have plotted it for $\wp ={10}^{-7}\pi$ in (a).

Figure 4

(Color online) Quantum escape rates ${\Gamma }_{\text{qm}}$ as a function of the bias current $\gamma$ for different values of $\wp$.

Figure 5

(Color online) Thermal escape rates ${\Gamma }_{\text{th}}$ as a function of the bias current $\gamma$ for different temperatures. Dots represent the quantum escape rate ${\Gamma }_{\text{qm}}$. All escape rates are calculated for $\wp =\pi$.