Quantum tunneling of semifluxons in a $0\text{−}\pi \text{−}0$ long Josephson junction

We consider a system of two semifluxons of opposite polarity in a $0\text{−}\pi \text{−}0$ long Josephson junction, which classically can be in one of two degenerate states: ↑↓ or ↓↑. When the distance $a$ between the $0\text{−}\pi$ boundaries (semifluxon’s centers) is a bit larger than the crossover distance $a_c$, the system can switch from one state to the other due to thermal fluctuations or quantum tunneling. We map this problem to the dynamics of a single particle in a double well potential and estimate parameters for which quantum effects emerge. We also determine the classical-to-quantum crossover temperature as well as the tunneling rate (energy level splitting) between the states ↑↓ and ↓↑.

Figure 1

(Color online) Two solutions ${\mu }_{\uparrow }\left(x\right)$ and ${\mu }_{\downarrow }\left(x\right)$ corresponding to the two different states ↑ and ↓ of a single semifluxon in a $0\text{−}\pi$ LJJ. The background color shows the corresponding Josephson potential energy density $\mathcal{U}=1-\cos (\mu +\theta )$, black corresponds to the valleys $\mathcal{U}=0$, while white corresponds to the summits $\mathcal{U}=2$.

Figure 2

(Color online) Two solutions ${\mu }_{\uparrow \downarrow }\left(x\right)$ and ${\mu }_{\downarrow \uparrow }\left(x\right)$ corresponding to the two different states ↑↓ and ↓↑ of two semifluxons in $0\text{−}\pi \text{−}0$ LJJ. The background color shows the corresponding Josephson potential energy $\mathcal{U}=1-\cos (\mu +\theta )$, black corresponds to the valleys $\mathcal{U}=0$, while white to the summits $\mathcal{U}=2$. The solid line shows the exact numerical solution of sine-Gordon Eq. (3), while the dashed line almost undistinguishable from the solid line, shows the approximate solution (6) for $a=1.7{\lambda }_J$ for $B=+B_0$ (state ↑↓).

Figure 3

(Color online) Schematic view on two classically stable states ↑↓ and ↓↑ and the corresponding mapping to a single particle moving along coordinate $B$ in a one-dimensional potential $U\left(B\right)$. Quantum mechanically one should speak about probability density ${\mid \psi \left(B\right)\mid }^2$ to find the particle at different locations (area plot).

Figure 4

The two lowest energy eigenvalues in a double well potential and the corresponding eigenfunctions. The ground state ${\psi }_0\left(B\right)$ is symmetric whereas the first excited state ${\psi }_1\left(B\right)$ is antisymmetric. The sum ${\psi }_{+}\left(B\right)$ and the difference ${\psi }_{-}\left(B\right)$ of these two energy eigenfunctions are localized in the right or in the left well, respectively.

Figure 5

The energy splitting $\delta \epsilon$ as a function of $\delta \tilde{a}$ for $\hslash {\omega }_p∕\left(E_J{\lambda }_J\right)=2.4\times {10}^{-3}$. The gray line is the energy splitting according to the semiclassical approximation Eq. (26), while symbols show $\delta \epsilon$ obtained by numerical solution of Eq. (15).

Figure 6

(Color online) Potential $U\left(\delta \tilde{a}\right)-U\left(0\right)$ (thick line), the two lowest eigenfunctions ${\psi }_{0,1}$ (solid and dashed lines), and the two corresponding lowest energy eigenvalues $E_{0,1}$ (thin horizontal solid and dashed lines) for (a) $\delta \tilde{a}=0$, (b) $\delta \tilde{a}=0.01$, and (c) $\delta \tilde{a}=0.02$.